An Innovative Method for Deterministic Multifactor Analysis Based on Chain Substitution Averaging

Abstract

The aims of this paper are to present the methodology, derived mathematical expressions for determining the individual factor influences and the adaptation for the conditions of dynamic deterministic factor analysis and the results of the application of the developed new method for deterministic factor analysis, called the averaged chain substitution method. After formulating the concept of the considered approach, all mathematical expressions used to create models containing up to four factor variables are presented and summarized. The scientific novelty of the study is in the obtained new equations for determining the individual factor influences by the method of averaged chain substitution and the method of analogy for five-factor additive or difference-multiplicative and for five-factor additive or difference-multifactor models with an additive or different part in the numerator of the factor model. The presented mathematical expressions accurately and unambiguously quantify the impact of individual factor influences for all types of factor models and thus significantly expand the applicability of the averaged chain substitution method in the theory and practice of financial-economic analysis. The proposed formalization and algorithmization of the evaluation process makes the method easy to apply by all economic and financial analysts for the purposes of deterministic factor analysis. The methodology was applied to perform a dynamic deterministic factor analysis of the total liquidity of Monbat AD and ELHIM-ISKRA AD for the period 2017–2021, based on the consolidated annual financial statements of the companies, available on the website of the Bulgarian Stock Exchange.

1. Introduction

Deterministic factor analysis (DFA) is a branch of economic analysis. Its aim is to directly, accurately, and unambiguously quantify the effect that the absolute changes of the participating factor variables (factors) exert on the absolute change of the performance (resultant) indicator in mathematically deterministic factor models.

The types of factor model depend on the type of mathematical dependence describing the relationship between the performance (resultant) indicator (P) and the participating factor variables ($a$, $b$, $c$, …) in a mathematically deterministic factor model.

In the practice of deterministic factor analysis, the following types of factor model are known:

• additive or different models—in them, the performance indicator is the sum or difference of the factor variables involved, namely: $P=a\pm b\pm c\pm \cdots ;$
• multiplicative models—here, the performance indicator is the product of the factor variables involved, namely: $P=a \ast b \ast \dots$;
• multiple model—here, the performance indicator is the quotient of the factor variables involved, namely: $P=\frac{a}{b}$;
• mixed (combined) models—these are combinations of additive or different, multiplicative or multiple factor models and can be as follows: multiplicative-multiple, additive or different-multiple, additive or different-multiplicative, and additive or different-multiplicative-multiple models.

In additive factor models, the absolute change of the performance indicator is equal to the sum of the absolute changes of the participating factor variables, i.e., the individual factor influence is equal to the absolute change of the corresponding factor variable. In different factor models, which are a special case of additive factor models, the factor influence corresponds to the absolute change of the factor variable, but the direction (sign) of influence needs to be considered. For those factor variables preceded by a minus in the factor model, the direction of influence is reversed, i.e., the factor influence is equal to minus the absolute factor change.

In the remaining types of factor models, the distribution of the absolute change of the performance indicator (ΔP) by factor variables is based on the developments of a number of researchers; namely, Jugenburg [1] theoretically substantiates and derives the third variant of the method of simple addition of an “indecomposable remainder”. A. Humal [2], V.I. Fedorova and Yu. Egorov [3] give a detailed description of the logarithmic method for DFA. A.D. Sheremet, G.G. Day and V.N. Shapovalov [4] theoretically substantiate the integral method for DFA in 1971. V.E. Adamov [5] derives and defends the first and second variants of the method of simple addition of an “indecomposable remainder” in 1977. S.S. Lipovetsky [6] proposes an approach for the distribution of the growth of the resultative indicator by factor variables through the variation analysis, which is not widely used. A.Y. Vaninsky [7] further developed the integral method for a wider range of factorial models. G. Foster [8], L.A. Bernstein, J.J. Wilde, K.R. Subramanyam [9] give the practical application of DFA methods to the conditions of Financial Statement Analysis in the Western literature. S.A. Ross, R.W. Westerfield, J.F. Jaffe [10], D.R. Emery, J.D. Finnerty, J.D. Stowe [11] apply DFA methods in corporate financial management. M.I. Bakanov and A.D. Sheremed [12] in 2001 derived more working formulas for determining individual factor influences by the integral method. S.V. Chebotarev [13] developed the method of economic factor analysis based on Lagrange mean value theorem. N.Sh. Kremer [14], N.P. Lyubushin [15], K.N. Lebedev [16], V.A. Prokofiev, V.V. Nosov T.V. Salomatina [17], and G.V. Savitskaya [18] summarize the shortcomings of DFA methods developed up to five years ago. During the period 2020–2023, Mitev [19,20,21,22,23] develops, approves and offers a new DFA method called the averaged chain substitution method.

In DFA, the following methods have been developed for quantifying the impact of the factors in mathematically deterministic factor models: the differential method; the coefficient method; the chain substitution method; the absolute difference method; the relative difference method; the shareholding method; the method of simple addition of an indecomposable remainder; the weighted finite difference method; the logarithmic method; the method of increment division into factors; the integral method; the index method.

Each of the DFA methods has its stages outlined, a specific application, limitations, capacities, advantages, and disadvantages. All these methods are presented in detail in the scientific and educational literature in the field of DFA. Unfortunately, none of the above methods achieves accuracy and unambiguity when it comes to the distribution of the so-called “indecomposable remainder” among the individual quantitative influences of factor variables on the absolute change of the performance indicator. This “indecomposable remainder” is the result of the simultaneous (combined) change of factor variables in the presence of a multiplicative or multiple (aliquot) element in the type of factor model. The various DFA methods employ different means of distributing the “indecomposable remainder” among the individual factor influences. Only the integral method achieves accuracy and unambiguity, but solely with multiplicative factor models. It allows for an arithmetic mean distribution of the “indecomposable remainder” among the individual factor influences.

The aims of this paper are to present the methodology, derived mathematical expressions for determining the individual factor influences and the adaptation for the conditions of dynamic deterministic factor analysis and the results of the application of the developed new method for deterministic factor analysis, called the averaged chain substitution method.

2. Applicability of Deterministic Factor Analysis Methods to the Types of Factor Model

If we summarize the possibilities for applying the various DFA methods according to the types of factor model based on the scientific literature, we can come up with the compilation in

Table 1. Possibilities of applying the methods of deterministic factor analysis by the individual types of factor models.

Methods of DFA	Types of Factor Models
	Additive or Different	Multiplicative	Multiple	Mixed
Differential method	−	+	−	−
Coefficient method	−	+	−	−
Chain substitution method	+	+	+	+
Index method	−	+	+	−
Absolute difference method	−	+	−	Only when $P=a\left(b-c\right)$
Absolute relative method	−	+	−	Only when $P=a\left(b-c\right)$
Equity participation method	−	+ Only when $\Delta P\ne 0$	−	−
The method of simple addition of an indecomposable remainder	−	+	−	−
Weighted finite difference method	−	+	−	−
The method of increment division into factors	−	+	−	−
Integral method	−	+	+	Only when $P=\frac{a}{b+\cdots +e}$
Logarithmic method	−	+ Only when ∆P ≠ 0	+ Only when  $\Delta P\ne 0$	Only when $P=\frac{a \ast \cdots \ast d}{e \ast \cdots \ast i}$ and  $\Delta P\ne 0$

Notes: +—the method is applicable. −—the method is inapplicable.

.

As can be seen from Table 1, some of the methods have been developed and are only applicable to multiplicative factor models. These are: the differential method; the coefficient method; the absolute difference method; the relative difference method; the equity participation method; the method of simple addition of an indecomposable remainder; the weighted finite difference method; the method of increment division into factors. This greatly restricts their application to the needs of the DFA.

Another part of the methods has been developed only for two-factor models. These are: the differential method; the method of simple addition of the indecomposable remainder; the weighted finite difference method; the method of increment division into factors, and the logarithmic method.

The equity participation method and the logarithmic method are applicable only when the change in the performance indicator ($\Delta P$) and, accordingly, the sum of the individual factor influences is different from zero.

In the practice of DFA, the methods that are employed most often are the integral method and the chain substitution method. The essence, stages, applicability, accuracy, advantages, and disadvantages of both methods are presented in detail in the scientific and educational literature in the field of DFA.

The chain substitution method has absolute universality of application. It is applicable to all possible types of factor model; however, it can not produce accurate and unambiguous results because the quantitative values of the influences of individual factors depend on the order of substitution of the factor variables during the construction of factor chains and the subsequent determining of individual factor influences. This is the only and undefinable disadvantage of the chain substitution method, namely: ambiguous results are obtained in the quantification of individual factor influences when changing the order of substitution of the factor variables during factor chain construction. This shortcoming brings about the need to rank (arrange) the factor variables. It is necessary to determine very precisely which of the factors involved in the factor model is primary, which is secondary, which is third in order, etc. However, the order is hard to justify, thus posing significant difficulties to managers or financial analysts. The chain substitution method generates the so-called stepwise (and, accordingly, uneven) distribution of the “indecomposable remainder”. The first factor is allocated no “indecomposable remainder”, whereas each successive factor is allocated a different portion of its distribution depending on the type of factor model.

The integral method was developed in 1971 by a group of Russian scientists, namely: Sheremet A.D., Djej G.G., and Shapovalov V.N. It was developed for a limited range of types of factor models, namely: for the multiplicative factor models ($P=a \ast b \ast \dots$), for a multiple factor model ($P=\frac{a}{b}$), and for a limited range of additive-multiple factor models of the type $P=\frac{a}{b+c+\cdots }$, where: P is the performance indicator in the factor model, and $a$, $b$, $c$,… are the factor variables involved in the factor model.

Russian scientists M.I. Bakanov and A.D. Sheremed [11] (pp. 139–143) have summarized working formulae for quantitative determination of the individual factor influences of a limited number of types of factor models, presented in

Table 2. Mathematical expressions for quantitative determination of individual factor influences by the integral method in different types of factor models [11] (pp. 139–143).

Factor Models, (Limitation)	$\mathbf{Factor} \mathbf{Influence} \mathit{a},$ $\mathbf{∆}{\mathit{P}}_{\left(\mathit{a}\right)}$	$\mathbf{Factor} \mathbf{Influence} \mathit{b},$ $\Delta {\mathit{P}}_{\left(\mathit{b}\right)}$	$\mathbf{Factor} \mathbf{Influence} \mathit{c},$ $\Delta {\mathit{P}}_{\left(\mathit{c}\right)}$	$\mathbf{Factor} \mathbf{Influence} \mathit{d},$ $\Delta {\mathit{P}}_{\left(\mathit{d}\right)}$
Multiplicative Factor Models
$P=a \ast b$	$\Delta P_A=b_0.\Delta a+\frac{\Delta a.\Delta b}{2}$	$\Delta P_B=a_0.\Delta b+\frac{\Delta a.\Delta b}{2}$	-	-
$P=a \ast b \ast c$	$\begin{array}{c}\Delta P_A=\frac{1}{2}\Delta a\left(b_0{c}_1+b_1{c}_0\right)\\ +\frac{1}{3}\Delta a.\Delta b.\Delta c\end{array}$	$\begin{array}{c}\Delta P_B=\frac{1}{2}\Delta b\left(a_0{c}_1+a_1{c}_0\right)\\ +\frac{1}{3}\Delta a.\Delta b.\Delta c\end{array}$	$\begin{array}{c}\Delta P_C=\frac{1}{2}\Delta c\left(a_0{b}_1+a_1{b}_0\right)\\ +\frac{1}{3}\Delta a.\Delta b.\Delta c\end{array}$	-
$P=a \ast b \ast c \ast d$	$\begin{array}{cc}\hfill \Delta P_A=& \frac{1}{6}\Delta a\left\{\begin{array}{c}3d_0{b}_0{c}_0+\\ +b_1{d}_0\left(c_1+\Delta c\right)+\\ +d_1{c}_0\left(b_1+\Delta b\right)+\\ +c_1{b}_0\left(d_1+\Delta d\right)\end{array}\right\}\hfill \\ & +\frac{1}{4}\Delta a.\Delta b.\Delta c.\Delta d\end{array}$	$\begin{array}{cc}\hfill \Delta P_B=& \frac{1}{6}\Delta b\left\{\begin{array}{c}3d_0{a}_0{c}_0+\\ +a_1{d}_0\left(c_1+\Delta c\right)+\\ +d_1{c}_0\left(a_1+\Delta a\right)+\\ +c_1{a}_0\left(d_1+\Delta d\right)\end{array}\right\}\hfill \\ & +\frac{1}{4}\Delta a.\Delta b.\Delta c.\Delta d\hfill \end{array}$	$\begin{array}{cc}\hfill \Delta P_c=& \frac{1}{6}\Delta c\left\{\begin{array}{c}3d_0{a}_0{b}_0+\\ +d_1{a}_0\left(b_1+\Delta b\right)+\\ +b_1{d}_0\left(a_1+\Delta a\right)+\\ +a_1{b}_0\left(d_1+\Delta d\right)\end{array}\right\}\hfill \\ & +\frac{1}{4}\Delta a.\Delta b.\Delta c.\Delta d\hfill \end{array}$	$\begin{array}{cc}\hfill \Delta P_d=& \frac{1}{6}\Delta d\left\{\begin{array}{c}3c_0{a}_0{b}_0+\\ +c_1{a}_0\left(b_1+\Delta b\right)+\\ +b_1{c}_0\left(a_1+\Delta a\right)+\\ +a_1{b}_0\left(c_1+\Delta c\right)\end{array}\right\}\hfill \\ & +\frac{1}{4}\Delta a.\Delta b.\Delta c.\Delta d\hfill \end{array}$
Additive-Multiple Factor Models
$$\begin{gathered} P=\frac{a}{b} \\ \left(b\ne 0\right) \end{gathered}$$	$\Delta P_A=\frac{\Delta a}{\Delta b}\mathit{ln}\left|\frac{b_1}{b_0}\right|$	$\Delta P_B=\Delta P-\Delta P_a$	-	-
$$\begin{gathered} P=\frac{a}{b+c} \\ \left(b+c\ne 0\right) \end{gathered}$$	$\Delta P_A=\frac{\Delta a}{\Delta b+\Delta c}\mathit{ln}\left|\frac{b_1+c_1}{b_0+c_0}\right|$	$\Delta P_B=\frac{\Delta P-\Delta P_a}{\Delta b+\Delta c}.\Delta b$	$\Delta P_C=\frac{\Delta P-\Delta P_a}{\Delta b+\Delta c}.\Delta c\hspace{0.33em}\mathrm{и}\mathsf{Л}\mathrm{и}$ $\Delta P_C=\Delta P-\left(\Delta P_a+\Delta P_b\right)$	-
$$\begin{gathered} P=\frac{a}{b+c+d} \\ \left(b+c+d\ne 0\right) \end{gathered}$$	$\Delta P_A=\frac{\Delta a}{\Delta b+\Delta c+\Delta d} \ast \mathit{ln}\left|\frac{b_1+c_1+d_1}{b_0+c_0+d_0}\right|$	$\Delta P_B=\frac{\Delta P-\Delta P_a}{\Delta b+\Delta c+\Delta d}.\Delta b$	$\Delta P_C=\frac{\Delta P-\Delta P_a}{\Delta b+\Delta c+\Delta d}.\Delta c$	$\Delta P_D=\frac{\Delta P-\Delta P_a}{\Delta b+\Delta c+\Delta d}.\Delta d$
$$\begin{gathered} P=\frac{a}{b+c+d+e} \\ \left(b+c+d+e\ne 0\right) \end{gathered}$$	$\Delta P_A=\frac{\Delta a}{\Delta b+\Delta c+\Delta d+\Delta e} \ast ln\left|\frac{b_1+c_1+d_1+e_1}{b_0+c_0+d_0+e_0}\right|$	$\Delta P_B=\frac{\Delta P-\Delta P_a}{b_0+c_0+d_0+e_0} \ast \Delta \mathrm{b}$	$\Delta P_C=\frac{\Delta P-\Delta P_a}{b_0+c_0+d_0+e_0} \ast \Delta \mathrm{c}$	$\Delta P_D=\frac{\Delta P-\Delta P_a}{b_0+c_0+d_0+e_0} \ast \Delta \mathrm{d}$ and $\Delta P_E=\frac{\Delta P-\Delta P_a}{b_0+c_0+d_0+e_0} \ast \Delta \mathrm{e}$

.

The stages, essence, advantages, disadvantages, and results of the approbation of the average method of chain substitutions are presented by Mitev [21,22]. Unlike the other DFA methods, the averaged chain substitution method has complete universality of application and absolute accuracy and clarity for all types of factor models when quantifying the individual factor influences of the involved factor variables on the absolute change of the performance indicator ($\Delta P$).

3. Research Methods and Methodology of the Averaged Chain Substitution Method

The following methods have been used in the research: critical analysis; synthesis; the dialectical approach; combinatorics; the method of mean values; the method of chain substitutions; and the average method of chain substitutions.

The stages of the averaged chain substitution method are presented in Figure 1.

The essence of the methodology of the averaged chain substitution method is based on the derivation of all mathematical expressions for determining the individual influence of each factor variable through the method of chain substitutions for each and every possible combination of the substitution sequence of the basic (planned) and the current (actual) values of the factor variables in the factor model analyzed. The number of possible combinations ($N$) is determined by the expression: $N=n!$, where n is the number of factor variables involved in the particular factor model. The obtained mathematical expressions for quantifying the individual influence of the first factor in each possible combination of the substitution sequence of the participating factors when constructing the factor chains through the chain substitution method are averaged by their summation and subsequent division of the number of possible combinations into the sequence of substitution of the factor variables ($N=n!$). The mathematical expression obtained for the quantitative individual influence of the first factor is subjected to mathematical transformations and cancellations until a simplified mathematical dependence is obtained for quantifying the individual factor influence on the absolute change of the performance indicator. This procedure is repeated for the second, the third, and every subsequent factor of the factor model analyzed.

Averaging the mathematical expressions obtained for determining the individual influence of each factor in the factor model by the method of chain substitutions for every possible combination in the order of the factor variable substitution when constructing the factor chains means that the probability of occurrence of each possible sequence of the order of substitution of the factor variables is the same. Here, we obtain a result which allows equal occurrence probability of every possible combination of the factor substitution sequence in the construction of the factor chains. It is not necessary to rank (arrange) the factors involved in the factor model, which is the only major disadvantage of the chain substitution method. As a result, a unique and accurate mathematical expression is obtained for direct quantification of the individual factor influence on the absolute change of the performance indicator of each of the participating factors in the factor model.

The assumption with the averaged chain substitution method is as follows: The analyzed period is considered discretely, i.e., in two moments $T_0$ and $T_1$ (beginning—basic or planning period, and end—reporting or current period). Accordingly, the change of factor variables during the period $T_0$ − $T_1$ takes place concurrently, i.e., the performance indicator ($P$ in the interval of its change ($P_0$ ⇾ $P_1$) alters at a constant rate, in other words, rectilinearly. This assumption is analogous to that of the integral method, of the third variant of the method of simple addition of the indecomposable remainder, and of the weighted finite difference method. All those methods result in an arithmetic mean distribution of the “indecomposable residual” to the factor variable influences.

The stages of the averaged chain substitution method with a two-factor multiplicative model of the type $P=a \ast b$, proceed in the following sequence:

(1)

The type of factor model is two-factor multiplicative;

(2)

The number of factor variables involved is two;

(3)

The number of possible combinations of the order of substitution of the factor variables is two ($N=n!=2 \ast 1=2$), namely: $a-b$ and $b-a$;

(4)

Construction of the factor chains after the method of chain substitutions in the order of substitution of factor variables $a-b$ in the factor chains, i.e., first $a$ then $b$. This is carried out according to the following expressions:

$$P_0=a_0 \ast b_0$$

(1)

$$P^{\prime }=a_1 \ast b_0$$

(2)

$$P_1=a_1 \ast b_1$$

(3)

(5)

Determining the influence of factor $a$ in a substitution of the type a − b in a substitution of the type:

$$\Delta P_{\left(a\right)}^{a-b}=P^{\prime }-P_0=a_1 \ast b_0-a_0 \ast b_0$$

(4)

(6)

Determining the influence of factor b in a substitution of the type a − b after this expression:

$$\Delta P_{\left(b\right)}^{a-b}=P_1-P^{\prime }=a_1 \ast b_1-a_1 \ast b_0$$

(5)

(7)

Construction of the factor chains after the method of chain substitutions in the order of substitution of factor variables $b-a$ в in the factor chains, i.e., first $b$ then $a$. This is carried out according to the following expressions:

$$P_0=a_0 \ast b_0$$

(6)

$$P^{\prime }=a_0 \ast b_1$$

(7)

$$P_1=a_1 \ast b_1$$

(8)

(8)

Determining the influence of factor $a$ in a substitution of the type $b-a$ following the expression below:

$$\Delta P_{\left(a\right)}^{b-a}=P^{\prime }-P_0=a_1 \ast b_1-a_0 \ast b_1$$

(9)

(9)

Determining the influence of factor $b$ in a substitution of the type $b-a$ after this expression:

$$\Delta P_{\left(b\right)}^{b-a}=P_1-P^{\prime }=a_0 \ast b_1-a_0 \ast b_0$$

(10)

(10)

Determining the averaged influence of factor $a$ after this expression:

$$\begin{array}{c}\hspace{1em}\Delta P_{\left(a\right)}=\frac{1}{N}\left(P_{\left(a\right)}^{a-b}+P_{\left(a\right)}^{b-a}\right)\hfill \\ \hspace{1em}=\frac{1}{2}\left(a_1 \ast b_0-a_0 \ast b_0+a_1 \ast b_1-a_0 \ast b_1\right)\hfill \\ =\frac{1}{2}\left(\Delta a \ast b_0+\Delta a \ast b_1\right)=\frac{\Delta a}{2}\left(b_0+b_1\right)\hfill \end{array}$$

(11)

(11)

Determining the averaged influence of factor $b$ after the expression:

$$\begin{array}{c}\hspace{1em}∆P_{\left(b\right)}=\frac{1}{N}\left(P_{\left(b\right)}^{a-b}+P_{\left(b\right)}^{b-a}\right)\hfill \\ \hspace{1em}=\frac{1}{2}\left(a_1 \ast b_1-a_1 \ast b_0+a_0 \ast b_1-a_0 \ast b_0\right)\hfill \\ =\frac{1}{2}\left(\Delta b \ast a_1+\Delta b \ast a_0\right)=\frac{\Delta b}{2}\left(a_1+a_0\right)\hfill \end{array}$$

(12)

Thus, for the individual factor influences, the following expressions are obtained:

• for factor $a$:

  $$\Delta P_{\left(a\right)}=\Delta a\frac{\left(b_0+b_1\right)}{2}$$

  (13)
• for factor $b$:

  $$\Delta P_{\left(b\right)}=\Delta b\frac{\left(a_0+a_1\right)}{2}$$

  (14)

The stages of the averaged chain substitution method with a two-factor multiple (aliquot) model of the type $P=\frac{a}{b}$, proceed in the following sequence:

(1)

The type of factor model is two-factor multiple;

(2)

The number of factor variables involved is two;

(3)

The number of possible combinations of the order of substitution of the factor variables is two ($N=n!=2 \ast 1=2$), namely: $a-b$ and $b-a$;

(4)

Construction of the factor chains after the method of chain substitutions in the order of substitution of factor variables $a-b$ in the factor chains, i.e., first $a$ then $b$. This is carried out according to the following expressions:

$$P_0=\frac{a_0}{b_0}$$

(15)

$$P^{\prime }=\frac{a_1}{b_0}$$

(16)

$$P_1=\frac{a_1}{b_1}$$

(17)

(5)

Determining the influence of factor $a$ in a substitution of the type $a-b$ following the expression:

$$\Delta P_{\left(a\right)}^{a-b}=\frac{a_1}{b_0}-\frac{a_0}{b_0}$$

(18)

(6)

Determining the influence of factor $b$ in a substitution of the type $a-b$ after this expression:

$$\Delta P_{\left(b\right)}^{a-b}=\frac{a_1}{b_1}-\frac{a_1}{b_0}$$

(19)

(7)

Construction of the factor chains after the method of chain substitutions in the order of substitution of factor variables $b-a$ in the factor chains, i.e., first $b$ then $a$. This is carried out according to the following expressions:

$$P_0=\frac{a_0}{b_0}$$

(20)

$$P^{\prime }=\frac{a_0}{b_1}$$

(21)

$$P_1=\frac{a_1}{b_1}$$

(22)

(8)

Determining the influence of factor $a$ in a substitution of the type $b-a$ following the expression below:

$$\Delta P_{\left(a\right)}^{b-a}=P^{\prime }-P_0=\frac{a_1}{b_1}-\frac{a_0}{b_1}$$

(23)

(9)

Determining the influence of factor $b$ in a substitution of the type $b-a$ after this expression:

$$\Delta P_{\left(b\right)}^{b-a}=P_1-P^{\prime }=\frac{a_0}{b_1}-\frac{a_0}{b_0}$$

(24)

(10)

Determining the averaged influence of factor $a$ after this expression:

$$\begin{array}{c}\Delta P_{\left(a\right)}=\frac{1}{N}\left(P_{\left(a\right)}^{a-b}+P_{\left(a\right)}^{b-a}\right)\\ =\frac{1}{2}\left(\frac{a_1}{b_0}-\frac{a_0}{b_0}+\frac{a_1}{b_1}-\frac{a_0}{b_1}\right)=\frac{1}{2}\left(\frac{a_1-a_0}{b_0}+\frac{a_1-a_0}{b_1}\right)\\ =\frac{1}{2} \left(\frac{∆a}{b_0}+\frac{∆a}{b_1}\right)=\frac{∆a}{2} \left(\frac{1}{b_0}+\frac{1}{b_1}\right)\end{array}$$

(25)

(11)

Determining the averaged influence of factor $b$ after the expression:

$$\begin{array}{c}\Delta P_{\left(b\right)}=\frac{1}{N}\left(P_{\left(b\right)}^{a-b}+P_{\left(b\right)}^{b-a}\right)=\\ =\frac{1}{2}\left(\frac{a_1}{b_1}-\frac{a_1}{b_0}+\frac{a_0}{b_1}-\frac{a_0}{b_0}\right)=\frac{1}{2}\left(\frac{a_1+a_0}{b_1}-\frac{a_1+a_0}{b_0}\right)=\\ =\frac{1}{2}\left(\frac{a_0+a_1}{b_1}-\frac{a_0+a_1}{b_0}\right)\end{array}$$

(26)

Thus, the following expressions are obtained for the factor influences:

• for factor $a$:

  $$\Delta P_{\left(a\right)}=\frac{\Delta a}{2} \left(\frac{1}{b_0}+\frac{1}{b_1}\right)$$

  (27)
• for factor $b$:

  $$\Delta P_{\left(b\right)}=\frac{1}{2}\left(\frac{a_0+a_1}{b_1}-\frac{a_0+a_1}{b_0}\right)$$

  (28)

By analogy, the quantitative factor influences of the involved factor variables are derived for the remaining types of factor models after the averaged chain substitution method. It should be reminded here that the number of possible combinations of the order of substitution of factor variables when constructing the factor chains is determined by the expression: $N=n!$, where n is the number of factors in the factor model. With three-factor models: $N=n!=3 \ast 2 \ast 1=6$, with four-factor models: $N=n!=4 \ast 3 \ast 2 \ast 1=24$, etc.

4. Systematization of the Results of the Approbation of the Averaged Chain Substitution Method

Table 3. Systematization of multiplicative and multiple factor models and formulae for determining the individual factor influences by the averaged chain substitution method.

Factor Models, (Limitation)	$\mathrm{Factor} \mathrm{Influence} \mathit{a},$ $\Delta {\mathit{P}}_{\left(\mathit{a}\right)}$	$\mathrm{Factor} \mathrm{Influence} \mathit{b},$ $\Delta {\mathit{P}}_{\left(\mathit{b}\right)}$	$\mathrm{Factor} \mathrm{Influence} \mathit{c},$ $\Delta {\mathit{P}}_{\left(\mathit{c}\right)}$	$\mathrm{Factor} \mathrm{Influence} \mathit{d},$ $\Delta {\mathit{P}}_{\left(\mathit{d}\right)}$
Multiplicative Factor Models
$P=a \ast b$	$\frac{\Delta a}{2}\left(b_0+b_1\right)$	$\frac{\Delta b}{2}\left(a_0+a_1\right)$	-	-
$P=a \ast b \ast c$	$\frac{\Delta a}{3}\left(b_0.c_0+b_1.c_1+\frac{b_1.c_0+b_0.c_1}{2}\right)$	$\frac{\Delta b}{3}\left(a_0.c_0+a_1.c_1+\frac{a_1{c}_0+a_0.c_1}{2}\right)$	$\frac{\Delta c}{3}\left(a_0.b_0+a_1.b_1+\frac{a_1.b_0+a_0.b_1}{2}\right)$	-
$P=a \ast b \ast c \ast d$	$$\begin{gathered} \frac{\Delta a}{4}\left(b_0{c}_0{d}_0+b_1{c}_1{d}_1\right)+ \\ \frac{\Delta a}{12}\left(\begin{array}{c}{b}_1{c}_0{d}_0+b_0{c}_1{d}_0+\\ b_0{c}_0{d}_1+b_1{c}_1{d}_0+\\ b_1{c}_0{d}_1+b_0{c}_1{d}_1\end{array}\right) \end{gathered}$$	$$\begin{gathered} \frac{\Delta b}{4}\left(a_0{c}_0{d}_0+a_1{c}_1{d}_1\right)+ \\ \frac{\Delta b}{12}\left(\begin{array}{c}{a}_1{c}_0{d}_0+a_0{c}_1{d}_0+\\ a_0{c}_0{d}_1+a_1{c}_1{d}_0+\\ a_1{c}_0{d}_1+a_0{c}_1{d}_1\end{array}\right) \end{gathered}$$	$$\begin{gathered} \frac{\Delta c}{4}\left(a_0{b}_0{d}_0+a_1{b}_1{d}_1\right)+ \\ \frac{\Delta c}{12}\left(\begin{array}{c}{a}_1{b}_0{d}_0+a_0{b}_1{d}_0+\\ a_0{b}_0{d}_1+a_1{b}_1{d}_0+\\ a_1{b}_0{d}_1+a_0{b}_1{d}_1\end{array}\right) \end{gathered}$$	$$\begin{gathered} \frac{\Delta d}{4}\left(a_0{b}_0{c}_0+a_1{b}_1{c}_1\right)+ \\ \frac{\Delta d}{12}\left(\begin{array}{c}{a}_1{b}_0{c}_0+a_0{b}_1{c}_0+\\ a_0{b}_0{c}_1+a_1{b}_1{c}_0+\\ a_1{b}_0{c}_1+a_0{b}_1{c}_1\end{array}\right) \end{gathered}$$
Multiple Factor Models
$$\begin{gathered} P=\frac{a}{b} \\ \left(b\ne 0\right) \end{gathered}$$	$\frac{\Delta a}{2}\left(\frac{1}{b_0}+\frac{1}{b_1}\right)$	$\frac{1}{2}\left(\frac{a_1+a_0}{b_1}-\frac{a_1+a_0}{b_0}\right)$	-	-
$P=\frac{a}{\frac{b}{c}}=\frac{a \ast c}{b}$ (b$\ne 0)$	$\frac{\Delta a}{6}\left(\frac{2c_0+c_1}{b_0}+\frac{2c_1+c_0}{b_1}\right)$	$\frac{1}{6}\left(\begin{array}{c}\frac{2\left(a_1{c}_1+a_0{c}_0\right)+a_1{c}_0+a_0{c}_1}{b_1}\\ -\frac{2\left(a_1{c}_1+a_0{c}_0\right)+a_1{c}_0+a_0{c}_1}{b_0}\end{array}\right)$	$\frac{\Delta c}{6}\left(\frac{2a_0+a_1}{b_0}+\frac{2a_1+a_0}{b_1}\right)$	-
$P=\frac{\frac{a}{b}}{c}=\frac{a}{b \ast c}$ $(\mathrm{b}\ne 0;\mathrm{c}\ne 0)$	$\frac{1}{6}\left(\frac{2\Delta a}{b_0{c}_0}+\frac{2\Delta a}{b_1{c}_1}+\frac{\Delta a}{b_1{c}_0}+\frac{\Delta a}{b_0{c}_1}\right)$	$\frac{1}{6}\left(\begin{array}{c}\frac{2a_1+a_0}{b_1{c}_1}-\frac{2a_0+a_1}{b_0{c}_0}\\ +\frac{2a_0+a_1}{b_1{c}_0}-\frac{2a_1+a_0}{b_0{c}_1}\end{array}\right)$	$\frac{1}{6}\left(\begin{array}{c}\frac{2a_1+a_0}{b_1{c}_1}-\frac{2a_0+a_1}{b_0{c}_0}\\ +\frac{2a_0+a_1}{b_0{c}_1}-\frac{2a_1+a_0}{b_1{c}_0}\end{array}\right)$	-
$P=\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a \ast d}{b \ast c}$ (b$\ne 0;\mathrm{c}\ne 0)$	$\frac{\Delta a}{12}\left(\begin{array}{c}\frac{3d_0+d_1}{b_0{c}_0}+\frac{3d_1+d_0}{b_1{c}_1}\\ +\frac{d_0+d_1}{b_0{c}_1}+\frac{d_0+d_1}{b_1{c}_0}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3a_1{d}_1+a_0{d}_0+a_1{d}_0+a_0{d}_1}{b_1{c}_1}\\ -\frac{3a_0{d}_0+a_1{d}_1+a_1{d}_0+a_0{d}_1}{b_0{c}_0}\\ +\frac{3a_0{d}_0+a_1{d}_1+a_1{d}_0+a_0{d}_1}{b_1{c}_0}\\ -\frac{3a_1{d}_1+a_0{d}_0+a_1{d}_0+a_0{d}_1}{b_0{c}_1}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3a_1{d}_1+a_0{d}_0+a_1{d}_0+a_0{d}_1}{b_1{c}_1}\\ -\frac{3a_0{d}_0+a_1{d}_1+a_1{d}_0+a_0{d}_1}{b_0{c}_0}\\ +\frac{3a_0{d}_0+a_1{d}_1+a_1{d}_0+a_0{d}_1}{b_0{c}_1}\\ -\frac{3a_1{d}_1+a_0{d}_0+a_1{d}_0+a_0{d}_1}{b_1{c}_0}\end{array}\right)$	$\frac{\Delta d}{12}\left(\begin{array}{c}\frac{3a_0+a_1}{b_0{c}_0}+\frac{3a_1+a_0}{b_1{c}_1}\\ +\frac{a_0+a_1}{b_0{c}_1}+\frac{a_0+a_1}{b_1{c}_0}\end{array}\right)$

,

Table 4. Systematization of additive or different-multiple models and formulae for determining the individual factor influences by the averaged chain substitution method.

Factor Models	$\mathrm{Factor} \mathrm{Influence} \mathit{a}$, Δ${\mathit{P}}_{\left(a\right)}$	$\mathrm{Factor} \mathrm{Influence} \mathit{b}$, Δ${\mathit{P}}_{\left(\mathit{b}\right)}$	$\mathrm{Factor} \mathrm{Influence} \mathit{c}$, Δ${\mathit{P}}_{\left(\mathit{c}\right)}$	$\mathrm{Factor} \mathrm{Influence} \mathit{d}$, Δ${\mathit{P}}_{\left(\mathit{d}\right)}$
Additive or Different-Multiplicative Factor Models
$P=a\left(b+c\right)$	$\frac{\Delta a}{2}\left(b_0+b_1+c_0+c_1\right)$	$\frac{\Delta b}{2}\left(a_0+a_1\right)$	$\frac{\Delta c}{2}\left(a_0+a_1\right)$	-
$P=a\left(b-c\right)$	$\frac{\Delta a}{2}\left(b_0+b_1-c_0-c_1\right)$	$\frac{\Delta b}{2}\left(a_0+a_1\right)$	$\frac{-\Delta c}{2}\left(a_0+a_1\right)$
$P=a\left(b+c+d\right)$	$\frac{\Delta a}{2}\left(b_0+b_1+c_0+c_1+d_0+d_1\right)$	$\frac{\Delta b}{2}\left(a_0+a_1\right)$	$\frac{\Delta c}{2}\left(a_0+a_1\right)$	$\frac{\Delta d}{2}\left(a_0+a_1\right)$
$P=a\left(b-c-d\right)$	$\frac{\Delta a}{2}\left(b_0+b_1-c_0-c_1-d_0-d_1\right)$	$\frac{\Delta b}{2}\left(a_0+a_1\right)$	$\frac{-\Delta c}{2}\left(a_0+a_1\right)$	$\frac{-\Delta d}{2}\left(a_0+a_1\right)$
$P=a\left(b+c-d\right)$	$\frac{\Delta a}{2}\left(b_0+b_1+c_0+c_1-d_0-d_1\right)$	$\frac{\Delta b}{2}\left(a_0+a_1\right)$	$\frac{\Delta c}{2}\left(a_0+a_1\right)$	$\frac{-\Delta d}{2}\left(a_0+a_1\right)$
$P=a\left(b-c+d\right)$	$\frac{\Delta a}{2}\left(b_0+b_1-c_0-c_1+d_0+d_1\right)$	$\frac{\Delta b}{2}\left(a_0+a_1\right)$	$\frac{-\Delta c}{2}\left(a_0+a_1\right)$	$\frac{\Delta d}{2}\left(a_0+a_1\right)$

,

Table 5. Systematization of multiplicative-multiple factor models and formulae for determining the individual factor influences by the averaged chain substitution method.

Factor Models, (Limitation)	$\mathrm{Factor} \mathrm{Influence} \mathit{a}$, $\Delta {\mathit{P}}_{\left(\mathit{a}\right)}$	$\mathrm{Factor} \mathrm{Influence} \mathit{b}$, $\Delta {\mathit{P}}_{\left(\mathit{b}\right)}$	$\mathrm{Factor} \mathrm{Influence} \mathit{c}$, $\Delta {\mathit{P}}_{\left(\mathit{c}\right)}$	$\mathrm{Factor} \mathrm{Influence} \mathit{d}$, $\Delta {\mathit{P}}_{\left(\mathit{d}\right)}$
Multiplicative-Multiple Factor Models
$P=\frac{a}{b \ast c}$ $\left(b\ne 0;c\ne 0\right)$	$\frac{\Delta a}{6}\left(\frac{2}{b_0{c}_0}+\frac{2}{b_1{c}_1}+\frac{1}{b_1{c}_0}+\frac{1}{b_0{c}_1}\right)$	$\frac{1}{6}\left(\begin{array}{c}\frac{2a_1+a_0}{b_1{c}_1}-\frac{2a_0+a_1}{b_0{c}_0}\\ +\frac{2a_0+a_1}{b_1{c}_0}-\frac{2a_1+a_0}{b_0{c}_1}\end{array}\right)$	$\frac{1}{6}\left(\begin{array}{c}\frac{2a_1+a_0}{b_1{c}_1}-\frac{2a_0+a_1}{b_0{c}_0}\\ +\frac{2a_0+a_1}{b_0{c}_1}-\frac{2a_1+a_0}{b_1{c}_0}\end{array}\right)$	-
$P=\frac{a \ast b}{c}$ $\left(c\ne 0\right)$	$\frac{\Delta a}{6}\left(\frac{2b_0+b_1}{c_0}+\frac{2b_1+b_0}{c_1}\right)$	$\frac{\Delta b}{6}\left(\frac{2a_0+a_1}{c_0}+\frac{2a_1+a_0}{c_1}\right)$	$\frac{2\left(a_1{b}_1+a_0{b}_0\right)+a_1{b}_0+a_0{b}_1}{6}\left(\frac{1}{c_1}-\frac{1}{c_0}\right)$	-
$P=\frac{a}{b \ast c \ast d}$ $(b\ne 0;c\ne 0;$ $d\ne 0)$	$\frac{\Delta a}{12}\left(\begin{array}{c}\frac{3}{b_0{c}_0{d}_0}+\frac{3}{b_1{c}_1{d}_1}\\ +\frac{1}{b_1{c}_0{d}_0}+\frac{1}{b_1{c}_1{d}_0}\\ +\frac{1}{b_0{c}_1{d}_1}+\frac{1}{b_0{c}_1{d}_0}\\ +\frac{1}{b_1{c}_0{d}_1}+\frac{1}{b_0{c}_0{d}_1}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3a_1+a_0}{b_1{c}_1{d}_1}-\frac{3a_1+a_0}{b_0.c_1.d_1}\\ +\frac{3a_0+a_1}{b_1{c}_0{d}_0}-\frac{3a_0+a_1}{b_0{c}_0{d}_0}\\ +\frac{a_0+a_1}{b_1{c}_1{d}_0}-\frac{a_0+a_1}{b_0{c}_1{d}_0}\\ +\frac{a_0+a_1}{b_1{c}_0{d}_1}-\frac{a_0+a_1}{b_0{c}_0{d}_1}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3a_1+a_0}{b_1{c}_1{d}_1}-\frac{3a_0+a_1}{b_0{c}_0{d}_0}\\ +\frac{3a_0+a_1}{b_0{c}_1{d}_0}-\frac{3a_1+a_0}{b_1{c}_0{d}_1}\\ +\frac{a_0+a_1}{b_1{c}_1{d}_0}-\frac{a_0+a_1}{b_1{c}_0{d}_0}\\ +\frac{a_0+a_1}{b_0{c}_1{d}_1}-\frac{a_0+a_1}{b_0{c}_0{d}_1}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3a_1+a_0}{b_1{c}_1{d}_1}-\frac{3a_0+a_1}{b_0{c}_0{d}_0}\\ +\frac{3a_0+a_1}{b_0{c}_0{d}_1}-\frac{3a_1+a_0}{b_1{c}_1{d}_0}\\ +\frac{a_0+a_1}{b_0{c}_1{d}_1}-\frac{a_0+a_1}{b_0{c}_1{d}_0}\\ +\frac{a_0+a_1}{b_1{c}_0{d}_1}-\frac{a_0+a_1}{b_1{c}_0{d}_0}\end{array}\right)$
$P=\frac{a \ast b \ast c}{d}$ $\left(d\ne 0\right)$	$\frac{\Delta a}{12}\left(\begin{array}{c}\frac{3b_0{c}_0+b_1{c}_0+b_0{c}_1+b_1{c}_1}{d_0}+\\ \frac{3b_1{c}_1+b_1{c}_0+b_0{c}_1+b_0{c}_0}{d_1}\end{array}\right)$	$\frac{\Delta b}{12}\left(\begin{array}{c}\frac{3a_0{c}_0+a_1{c}_0+a_0{c}_1+a_1{c}_1}{d_0}+\\ \frac{3a_1{c}_1+a_1{c}_0+a_0{c}_1+a_0{c}_0}{d_1}\end{array}\right)$	$\frac{\Delta c}{12}\left(\begin{array}{c}\frac{3a_0{b}_0+a_1{b}_0+a_0{b}_1+a_1{b}_1}{d_0}+\\ \frac{3a_1{b}_1+a_1{b}_0+a_0{b}_1+a_0{b}_0}{d_1}\end{array}\right)$	$\frac{\begin{array}{c}3\left(a_0{b}_0{c}_0+a_1{b}_1{c}_1\right)\\ +a_1{b}_1{c}_0+a_1{b}_0{c}_1\\ +a_1{b}_0{c}_0+a_0{b}_1{c}_1\\ a_0{b}_1{c}_0+a_0{b}_0{c}_1\end{array}}{12}\left(\frac{1}{d_1}+\frac{1}{d_0}\right)$
$P=\frac{a \ast b}{c \ast d}$ $(c\ne 0;d\ne 0)$	$\frac{\Delta a}{12}\left(\begin{array}{c}\frac{3b_0+b_1}{c_0{d}_0}+\frac{3b_1+b_0}{c_1{d}_1}\\ +\frac{b_0+b_1}{c_0{d}_1}+\frac{b_0+b_1}{c_1{d}_0}\end{array}\right)$	$\frac{\Delta b}{12}\left(\begin{array}{c}\frac{3a_0+a_1}{c_0{d}_0}+\frac{3a_1+a_0}{c_1{d}_1}\\ +\frac{a_0+a_1}{c_0{d}_1}+\frac{a_0+a_1}{c_1{d}_0}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3a_1{b}_1+a_0{b}_0+a_1{b}_0+a_0{b}_1}{c_1{d}_1}\\ -\frac{3a_0{b}_0+a_1{b}_1+a_1{b}_0+a_0{b}_1}{c_0{d}_0}\\ +\frac{3a_0{b}_0+a_1{b}_1+a_1{b}_0+a_0{b}_1}{c_1{d}_0}\\ -\frac{3a_1{b}_1+a_0{b}_0+a_1{b}_0+a_0{b}_1}{c_0{d}_1}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3a_1{b}_1+a_0{b}_0+a_1{b}_0+a_0{b}_1}{c_1{d}_1}\\ -\frac{3a_0{b}_0+a_1{b}_1+a_1{b}_0+a_0{b}_1}{c_0{d}_0}\\ +\frac{3a_0{b}_0+a_1{b}_1+a_1{b}_0+a_0{b}_1}{c_0{d}_1}\\ -\frac{3a_1{b}_1+a_0{b}_0+a_1{b}_0+a_0{b}_1}{c_1{d}_0}\end{array}\right)$

,

Table 6. Systematization of additive or different-multiple factor models and formulae for determining the individual factor influences by the averaged chain substitution method.

Factor Models, (Limitation)	$\mathbf{Factor} \mathbf{Influence} \mathit{a}$, $\Delta {\mathit{P}}_{\left(\mathit{a}\right)}$	$\mathbf{Factor} \mathbf{Influence} \mathit{b},$ $\Delta {\mathit{P}}_{\left(\mathit{b}\right)}$	$\mathbf{Factor} \mathbf{Influence} \mathit{c},$ $\Delta {\mathit{P}}_{\left(\mathit{c}\right)}$	$\mathbf{Factor} \mathbf{Influence} \mathit{d},$ $\Delta {\mathit{P}}_{\left(\mathit{d}\right)}$
Additive or Different-Multiple Factor Models
$P=\frac{a}{b+c}$ ($b+c\ne 0)$	$\frac{\Delta a}{6}\left(\begin{array}{c}\frac{2}{b_0+c_0}+\frac{2}{b_1+c_1}+\\ \frac{1}{b_1+c_0}+\frac{1}{b_0+c_1}\end{array}\right)$	$\frac{1}{6}\left(\begin{array}{c}\frac{2a_1+a_0}{b_1+c_1}+\frac{2a_0+a_1}{b_1+c_0}-\\ \frac{2a_1+a_0}{b_0+c_1}-\frac{2a_0+a_1}{b_0+c_0}\end{array}\right)$	$\frac{1}{6}\left(\begin{array}{c}\frac{2a_1+a_0}{b_1+c_1}+\frac{2a_0+a_1}{b_0+c_1}-\\ \frac{2a_1+a_0}{b_1+c_0}-\frac{2a_0+a_1}{b_0+c_0}\end{array}\right)$	-
$P=\frac{a}{b-c}$ $(b-c\ne 0)$	$\frac{\Delta a}{6}\left(\begin{array}{c}\frac{2}{b_0-c_0}+\frac{2}{b_1-c_1}+\\ \frac{1}{b_1-c_0}+\frac{1}{b_0-c_1}\end{array}\right)$	$\frac{1}{6}\left(\begin{array}{c}\frac{2a_1+a_0}{b_1-c_1}+\frac{2a_0+a_1}{b_1-c_0}-\\ \frac{2a_1+a_0}{b_0-c_1}-\frac{2a_0+a_1}{b_0-c_0}\end{array}\right)$	$\frac{1}{6}\left(\begin{array}{c}\frac{2a_1+a_0}{b_1-c_1}+\frac{2a_0+a_1}{b_0-c_1}-\\ \frac{2a_1+a_0}{b_1-c_0}-\frac{2a_0+a_1}{b_0-c_0}\end{array}\right)$	-
$P=\frac{a+b}{c}$ $(\mathrm{c}\ne 0)$	$\frac{\Delta a}{2}\left(\frac{1}{c_0}+\frac{1}{c_1}\right)$	$\frac{1}{2}\left(\frac{\Delta b}{c_0}+\frac{\Delta b}{c_1}\right)$	$\frac{a_1+a_0+b_1+b_0}{2}\left(\begin{array}{c}\frac{1}{c_1}-\\ \frac{1}{c_0}\end{array}\right)$	-
$P=\frac{a-b}{c}$ (c$\ne 0)$	$\frac{\Delta a}{2}\left(\frac{1}{c_0}+\frac{1}{c_1}\right)$	$\frac{1}{2}\left(\frac{b_0-b_1}{c_0}+\frac{b_0-b_1}{c_1}\right)$	$\frac{a_1+a_0-b_1-b_0}{2}\left(\begin{array}{c}\frac{1}{c_1}+\\ \frac{1}{c_0}\end{array}\right)$	-
$P=\frac{a+b}{c+d}$ $\left(c+d\ne 0\right)$	$\frac{\Delta a}{6}\left(\begin{array}{c}\frac{2}{c_0+d_0}+\frac{2}{c_1+d_1}\\ +\frac{1}{c_0+d_1}+\frac{1}{c_1+d_0}\end{array}\right)$	$\frac{1}{6}\left(\begin{array}{c}\frac{2\Delta b}{c_0+d_0}+\frac{2\Delta b}{c_1+d_1}\\ +\frac{\Delta b}{c_0+d_1}+\frac{\Delta b}{c_1+d_0}\end{array}\right)$	$\frac{1}{6}\left[\begin{array}{c}\frac{2\left(a_1+b_1\right)+\left(a_0+b_0\right)}{c_1+d_1}\\ -\frac{2\left(a_0+b_0\right)+\left(a_1+b_1\right)}{c_0+d_0}\\ +\frac{2\left(a_0+b_0\right)+\left(a_1+b_1\right)}{c_1+d_0}\\ -\frac{2\left(a_1+b_1\right)+\left(a_0+b_0\right)}{c_0+d_1}\end{array}\right]$	$\frac{1}{6}\left[\begin{array}{c}\frac{2\left(a_1+b_1\right)+\left(a_0+b_0\right)}{c_1+d_1}\\ -\frac{2\left(a_0+b_0\right)+\left(a_1+b_1\right)}{c_0+d_0}\\ +\frac{2\left(a_0+b_0\right)+\left(a_1+b_1\right)}{c_0+d_1}\\ -\frac{2\left(a_1+b_1\right)+\left(a_0+b_0\right)}{c_1+d_0}\end{array}\right]$
$P=\frac{a-b}{c-d}$ $\left(c-d\ne 0\right)$	$\frac{\Delta a}{6}\left(\begin{array}{c}\frac{2}{c_0-d_0}+\frac{2}{c_1-d_1}\\ +\frac{1}{c_0-d_1}+\frac{1}{c_1-d_0}\end{array}\right)$	$-\frac{\Delta b}{6}\left(\begin{array}{c}\frac{2}{c_0-d_0}+\frac{2}{c_1-d_1}\\ +\frac{1}{c_0-d_1}+\frac{1}{c_1-d_0}\end{array}\right)$	$\frac{1}{6}\left[\begin{array}{c}\frac{2\left(a_1-b_1\right)+\left(a_0-b_0\right)}{c_1-d_1}\\ -\frac{2\left(a_0-b_0\right)+\left(a_1-b_1\right)}{c_0-d_0}\\ +\frac{2\left(a_0-b_0\right)+\left(a_1-b_1\right)}{c_1-d_0}\\ -\frac{2\left(a_1-b_1\right)+\left(a_0-b_0\right)}{c_0-d_1}\end{array}\right]$	$\frac{1}{6}\left[\begin{array}{c}\frac{2\left(a_1-b_1\right)+\left(a_0-b_0\right)}{c_1-d_1}\\ -\frac{2\left(a_0-b_0\right)+\left(a_1-b_1\right)}{c_0-d_0}\\ +\frac{2\left(a_0-b_0\right)+\left(a_1-b_1\right)}{c_0-d_1}\\ -\frac{2\left(a_1-b_1\right)+\left(a_0-b_0\right)}{c_1-d_0}\end{array}\right]$
$P=\frac{a+b}{c-d}$ $(c-d\ne 0)$	$\frac{\Delta a}{6}\left(\begin{array}{c}\frac{2}{c_0-d_0}+\frac{2}{c_1-d_1}\\ +\frac{1}{c_0-d_1}+\frac{1}{c_1-d_0}\end{array}\right)$	$\frac{\Delta b}{6}\left(\begin{array}{c}\frac{2}{c_0-d_0}+\frac{2}{c_1-d_1}\\ +\frac{1}{c_0-d_1}+\frac{1}{c_1-d_0}\end{array}\right)$	$\frac{1}{6}\left[\begin{array}{c}\frac{2\left(a_1+b_1\right)+\left(a_0+b_0\right)}{c_1+d_1}\\ -\frac{2\left(a_0+b_0\right)+\left(a_1+b_1\right)}{c_0+d_0}\\ +\frac{2\left(a_0+b_0\right)+\left(a_1+b_1\right)}{c_1+d_0}\\ -\frac{2\left(a_1+b_1\right)+\left(a_0+b_0\right)}{c_0+d_1}\end{array}\right]$	$\frac{1}{6}\left[\begin{array}{c}\frac{2\left(a_1+b_1\right)+\left(a_0+b_0\right)}{c_1-d_1}\\ -\frac{2\left(a_0+b_0\right)+\left(a_1+b_1\right)}{c_0+d_0}\\ +\frac{2\left(a_0+b_0\right)+\left(a_1+b_1\right)}{c_0-d_1}\\ -\frac{2\left(a_1+b_1\right)+\left(a_0+b_0\right)}{c_1-d_0}\end{array}\right]$
$P=\frac{a-b}{c+d}$ $(c+d\ne 0)$	$\frac{\Delta a}{6}\left(\begin{array}{c}\frac{2}{c_0+d_0}+\frac{2}{c_1+d_1}\\ +\frac{1}{c_0+d_1}+\frac{1}{c_1+d_0}\end{array}\right)$	$\frac{-\Delta b}{6}\left(\begin{array}{c}\frac{2}{c_0+d_0}+\frac{2}{c_1+d_1}\\ +\frac{1}{c_0+d_1}+\frac{1}{c_1+d_0}\end{array}\right)$	$\frac{1}{6}\left[\begin{array}{c}\frac{2\left(a_1-b_1\right)+\left(a_0-b_0\right)}{c_1-d_1}\\ -\frac{2\left(a_0-b_0\right)+\left(a_1-b_1\right)}{c_0-d_0}\\ +\frac{2\left(a_0-b_0\right)+\left(a_1-b_1\right)}{c_1-d_0}\\ -\frac{2\left(a_1-b_1\right)+\left(a_0-b_0\right)}{c_0-d_1}\end{array}\right]$	$\frac{1}{6}\left[\begin{array}{c}\frac{2\left(a_1-b_1\right)+\left(a_0-b_0\right)}{c_1+d_1}\\ -\frac{2\left(a_0-b_0\right)+\left(a_1-b_1\right)}{c_0+d_0}\\ +\frac{2\left(a_0-b_0\right)+\left(a_1-b_1\right)}{c_0+d_1}\\ -\frac{2\left(a_1-b_1\right)+\left(a_0-b_0\right)}{c_1+d_0}\end{array}\right]$
$P=\frac{a}{b+c+d}$ $\left(b+c+d\ne 0\right)$	$\frac{\Delta a}{12}\left(\begin{array}{c}\frac{3}{b_0+c_0+d_0}+\frac{3}{b_1+c_1+d_1}\\ +\frac{1}{b_1+c_0+d_0}+\frac{1}{b_1+c_1+d_0}\\ +\frac{1}{b_0+c_1+d_1}+\frac{1}{b_0+c_1+d_0}\\ +\frac{1}{b_0+c_0+d_1}+\frac{1}{b_1+c_0+d_1}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3.a_1+a_0}{b_1+c_1+d_1}-\frac{3.a_1+a_0}{b_0+c_1+d_1}\\ +\frac{3.a_0+a_1}{b_1+c_0+d_0}-\frac{3.a_0+a_1}{b_0+c_0+d_0}\\ +\frac{a_0+a_1}{b_1+c_1+d_0}-\frac{a_0+a_1}{b_0+c_1+d_0}\\ +\frac{a_0+a_1}{b_1+c_0+d_1}-\frac{a_0+a_1}{b_0+c_0+d_1}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3.a_1+a_0}{b_1+c_1+d_1}-\frac{3.a_0+a_1}{b_0+c_0+d_0}\\ +\frac{3.a_0+a_1}{b_0+c_1+d_0}-\frac{3.a_1+a_0}{b_1+c_0+d_0}\\ +\frac{a_0+a_1}{b_1+c_1+d_0}-\frac{a_0+a_1}{b_1+c_0+d_0}\\ +\frac{a_0+a_1}{b_0+c_1+d_1}-\frac{a_0+a_1}{b_0+c_0+d_1}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3.a_1+a_0}{b_1+c_1+d_1}-\frac{3.a_0+a_1}{b_0+c_0+d_0}\\ +\frac{3.a_0+a_1}{b_0+c_0+d_1}-\frac{3.a_1+a_0}{b_1+c_1+d_0}\\ +\frac{a_0+a_1}{b_0+c_1+d_1}-\frac{a_0+a_1}{b_0+c_1+d_0}\\ +\frac{a_0+a_1}{b_1+c_0+d_1}-\frac{a_0+a_1}{b_1+c_0+d_0}\end{array}\right)$
$P=\frac{a}{b-c-d}$ $\left(b-c-d\ne 0\right)$	$\frac{\Delta a}{12}\left(\begin{array}{c}\frac{3}{b_0-c_0-d_0}+\frac{3}{b_1-c_1-d_1}\\ +\frac{1}{b_1-c_0-d_0}+\frac{1}{b_1-c_1-d_0}\\ +\frac{1}{b_0-c_1-d_1}+\frac{1}{b_0-c_1-d_0}\\ +\frac{1}{b_0-c_0-d_1}+\frac{1}{b_1-c_0-d_1}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3.a_1+a_0}{b_1-c_1-d_1}-\frac{3.a_1+a_0}{b_0-c_1-d_1}\\ +\frac{3.a_0+a_1}{b_1-c_0-d_0}-\frac{3.a_0+a_1}{b_0-c_0-d_0}\\ +\frac{a_0+a_1}{b_1-c_1-d_0}-\frac{a_0+a_1}{b_0-c_1-d_0}\\ +\frac{a_0+a_1}{b_1-c_0-d_1}-\frac{a_0+a_1}{b_0-c_0-d_1}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3.a_1+a_0}{b_1-c_1-d_1}-\frac{3.a_0+a_1}{b_0-c_0-d_0}\\ +\frac{3.a_0+a_1}{b_0-c_1-d_0}-\frac{3.a_1+a_0}{b_1-c_0-d_0}\\ +\frac{a_0+a_1}{b_1-c_1-d_0}-\frac{a_0+a_1}{b_1-c_0-d_0}\\ +\frac{a_0+a_1}{b_0-c_1-d_1}-\frac{a_0+a_1}{b_0-c_0-d_1}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3.a_1+a_0}{b_1-c_1-d_1}-\frac{3.a_0+a_1}{b_0-c_0-d_0}\\ +\frac{3.a_0+a_1}{b_0-c_0-d_1}-\frac{3.a_1+a_0}{b_1-c_1-d_0}\\ +\frac{a_0+a_1}{b_0-c_1-d_1}-\frac{a_0+a_1}{b_0-c_1-d_0}\\ +\frac{a_0+a_1}{b_1-c_0-d_1}-\frac{a_0+a_1}{b_1-c_0-d_0}\end{array}\right)$
$P=\frac{a}{b-c+d}$ $\left(b-c+d\ne 0\right)$	$\frac{\Delta a}{12}\left(\begin{array}{c}\frac{3}{b_0-c_0+d_0}+\frac{3}{b_1-c_1+d_1}\\ +\frac{1}{b_1-c_0+d_0}+\frac{1}{b_1-c_1+d_0}\\ +\frac{1}{b_0-c_1+d_1}+\frac{1}{b_0-c_1+d_0}\\ +\frac{1}{b_0-c_0+d_1}+\frac{1}{b_1-c_0+d_1}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3.a_1+a_0}{b_1-c_1+d_1}-\frac{3.a_1+a_0}{b_0-c_1+d_1}\\ +\frac{3.a_0+a_1}{b_1-c_0+d_0}-\frac{3.a_0+a_1}{b_0-c_0+d_0}\\ +\frac{a_0+a_1}{b_1-c_1+d_0}-\frac{a_0+a_1}{b_0-c_1+d_0}\\ +\frac{a_0+a_1}{b_1-c_0-d_1}-\frac{a_0+a_1}{b_0-c_0+d_1}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3.a_1+a_0}{b_1-c_1+d_1}-\frac{3.a_0+a_1}{b_0-c_0+d_0}\\ +\frac{3.a_0+a_1}{b_0-c_1+d_0}-\frac{3.a_1+a_0}{b_1-c_0+d_0}\\ +\frac{a_0+a_1}{b_1-c_1+d_0}-\frac{a_0+a_1}{b_1-c_0+d_0}\\ +\frac{a_0+a_1}{b_0-c_1+d_1}-\frac{a_0+a_1}{b_0-c_0+d_1}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3.a_1+a_0}{b_1-c_1+d_1}-\frac{3.a_0+a_1}{b_0-c_0+d_0}\\ +\frac{3.a_0+a_1}{b_0-c_0+d_1}-\frac{3.a_1+a_0}{b_1-c_1+d_0}\\ +\frac{a_0+a_1}{b_0-c_1+d_1}-\frac{a_0+a_1}{b_0-c_1+d_0}\\ +\frac{a_0+a_1}{b_1-c_0+d_1}-\frac{a_0+a_1}{b_1-c_0+d_0}\end{array}\right)$
$P=\frac{a}{b+c-d}$ $\left(b+c-d\ne 0\right)$	$\frac{\Delta a}{12}\left(\begin{array}{c}\frac{3}{b_0+c_0-d_0}+\frac{3}{b_1+c_1-d_1}\\ +\frac{1}{b_1+c_0-d_0}+\frac{1}{b_1+c_1-d_0}\\ +\frac{1}{b_0+c_1-d_1}+\frac{1}{b_0+c_1-d_0}\\ +\frac{1}{b_0+c_0-d_1}+\frac{1}{b_1+c_0-d_1}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3.a_1+a_0}{b_1+c_1-d_1}-\frac{3.a_1+a_0}{b_0+c_1-d_1}\\ +\frac{3.a_0+a_1}{b_1+c_0-d_0}-\frac{3.a_0+a_1}{b_0+c_0-d_0}\\ +\frac{a_0+a_1}{b_1+c_1-d_0}-\frac{a_0+a_1}{b_0+c_1-d_0}\\ +\frac{a_0+a_1}{b_1+c_0-d_1}-\frac{a_0+a_1}{b_0+c_0-d_1}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3.a_1+a_0}{b_1+c_1-d_1}-\frac{3.a_0+a_1}{b_0+c_0-d_0}\\ +\frac{3.a_0+a_1}{b_0+c_1-d_0}-\frac{3.a_1+a_0}{b_1+c_0-d_0}\\ +\frac{a_0+a_1}{b_1+c_1-d_0}-\frac{a_0+a_1}{b_1+c_0-d_0}\\ +\frac{a_0+a_1}{b_0+c_1-d_1}-\frac{a_0+a_1}{b_0+c_0-d_1}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3.a_1+a_0}{b_1+c_1-d_1}-\frac{3.a_0+a_1}{b_0+c_0-d_0}\\ +\frac{3.a_0+a_1}{b_0+c_0-d_1}-\frac{3.a_1+a_0}{b_1+c_1-d_0}\\ +\frac{a_0+a_1}{b_0+c_1-d_1}-\frac{a_0+a_1}{b_0+c_1-d_0}\\ +\frac{a_0+a_1}{b_1+c_0-d_1}-\frac{a_0+a_1}{b_1+c_0-d_0}\end{array}\right)$
$P=\frac{a+b+c}{d}$ $\left(d\ne 0\right)$	$\frac{\Delta a}{2}\left(\frac{1}{d_0}+\frac{1}{d_1}\right)$	$\frac{\Delta b}{2}\left(\frac{1}{d_0}+\frac{1}{d_1}\right)$	$\frac{\Delta c}{2}\left(\frac{1}{d_0}+\frac{1}{d_1}\right)$	$\frac{a_0+a_1+b_0+b_1+c_0+c_1}{2}\left(\frac{1}{d_1}-\frac{1}{d_0}\right)$
$P=\frac{a-b-c}{d}$ $\left(d\ne 0\right)$	$\frac{\Delta a}{2}\left(\frac{1}{d_0}+\frac{1}{d_1}\right)$	$\frac{-\Delta b}{2}\left(\frac{1}{d_0}+\frac{1}{d_1}\right)$	$\frac{-\Delta c}{2}\left(\frac{1}{d_0}+\frac{1}{d_1}\right)$	$\frac{a_0+a_1-b_0-b_1-c_0-c_1}{2}\left(\frac{1}{d_1}-\frac{1}{d_0}\right)$
$P=\frac{a+b-c}{d}$ $\left(d\ne 0\right)$	$\frac{\Delta a}{2}\left(\frac{1}{d_0}+\frac{1}{d_1}\right)$	$\frac{\Delta b}{2}\left(\frac{1}{d_0}+\frac{1}{d_1}\right)$	$\frac{-\Delta c}{2}\left(\frac{1}{d_0}+\frac{1}{d_1}\right)$	$\frac{a_0+a_1+b_0+b_1-c_0-c_1}{2}\left(\frac{1}{d_1}-\frac{1}{d_0}\right)$
$P=\frac{a-b+c}{d}$ $\left(d\ne 0\right)$	$\frac{\Delta a}{2}\left(\frac{1}{d_0}+\frac{1}{d_1}\right)$	$\frac{-\Delta b}{2}\left(\frac{1}{d_0}+\frac{1}{d_1}\right)$	$\frac{\Delta c}{2}\left(\frac{1}{d_0}+\frac{1}{d_1}\right)$	$\frac{a_0+a_1-b_0-b_1+c_0+c_1}{2}\left(\frac{1}{d_1}-\frac{1}{d_0}\right)$

and

Table 7. Systematization of additive or different-multiplicative-multiple factor models and formulae of for determining the individual factor influences by the averaged chain substitution method.

Factor Models, (Limitation)	$\mathbf{Factor} \mathbf{Influence} \mathit{a},$ $\Delta {\mathit{P}}_{\left(\mathit{a}\right)}$	$\mathbf{Factor} \mathbf{Influence} \mathit{b},$ $\Delta {\mathit{P}}_{\left(\mathit{b}\right)}$	$\mathbf{Factor} \mathbf{Influence} \mathit{c},$ $\Delta {\mathit{P}}_{\left(\mathit{c}\right)}$	$\mathbf{Factor} \mathbf{Influence} \mathit{d},$ $\Delta {\mathit{P}}_{\left(\mathit{d}\right)}$
Additive or Different-Multiplicative-Multiple Factor Models
$P=\frac{a+b}{c \ast d}$ $\left(c\ne 0; d\ne 0\right)$	$\frac{\Delta a}{6}\left(\frac{2}{c_0.d_0}+\frac{2}{c_1.d_1}+\frac{1}{c_0.d_1}+\frac{1}{c_1.d_0}\right)$	$\frac{\Delta b}{6}\left(\frac{2}{c_0.d_0}+\frac{2}{c_1.d_1}+\frac{1}{c_0.d_1}+\frac{1}{c_1.d_0}\right)$	$\frac{1}{6}\left[\begin{array}{c}\frac{2\left(a_1+b_1\right)+a_0+b_0}{c_1.d_1}\\ -\frac{2\left(a_0+b_0\right)+a_1+b_1}{c_0.d_0}\\ +\frac{2\left(a_0+b_0\right)+a_1+b_1}{c_0.d_0}\\ -\frac{2\left(a_1+b_1\right)+a_0+b_0}{c_0.d_1}\end{array}\right]$	$\frac{1}{6}\left[\begin{array}{c}\frac{2\left(a_1+b_1\right)+a_0+b_0}{c_1.d_1}\\ -\frac{2\left(a_0+b_0\right)+a_1+b_1}{c_0.d_0}\\ +\frac{2\left(a_0+b_0\right)+a_1+b_1}{c_1.d_0}\\ -\frac{2\left(a_1+b_1\right)+a_0+b_0}{c_0.d_1}\end{array}\right]$
$P=\frac{a-b}{c \ast d}$ ($\mathrm{c}\ne 0; \mathrm{d}\ne 0)$	$\frac{\Delta a}{6}\left(\frac{2}{c_0.d_0}+\frac{2}{c_1.d_1}+\frac{1}{c_0.d_1}+\frac{1}{c_1.d_0}\right)$	$\frac{-\Delta b}{6}\left(\frac{2}{c_0.d_0}+\frac{2}{c_1.d_1}+\frac{1}{c_0.d_1}+\frac{1}{c_1.d_0}\right)$	$\frac{1}{6}\left[\begin{array}{c}\frac{2\left(a_1-b_1\right)+a_0-b_0}{c_1.d_1}\\ -\frac{2\left(a_0-b_0\right)+a_1-b_1}{c_0.d_0}\\ +\frac{2\left(a_0-b_0\right)+a_1-b_1}{c_1.d_0}\\ -\frac{2\left(a_1-b_1\right)+a_0-b_0}{c_0.d_1}\end{array}\right]$	$\frac{1}{6}\left[\begin{array}{c}\frac{2\left(a_1-b_1\right)+a_0-b_0}{c_1.d_1}\\ -\frac{2\left(a_0-b_0\right)+a_1-b_1}{c_0.d_0}\\ +\frac{2\left(a_0-b_0\right)+a_1-b_1}{c_0.d_1}\\ -\frac{2\left(a_1-b_1\right)+a_0-b_0}{c_1.d_0}\end{array}\right]$
$P=\frac{a \ast b}{c+d}$ ($\mathrm{c}+\mathrm{d}\ne 0)$	$\frac{\Delta a}{12}\left(\begin{array}{c}\frac{3.b_0+b_1}{c_0+d_0}+\frac{3.b_1+b_0}{c_1+d_1}\\ +\frac{b_0+b_1}{c_0+d_1}+\frac{b_0+b_1}{c_1+d_0}\end{array}\right)$	$\frac{\Delta b}{12}\left(\begin{array}{c}\frac{3.a_0+a_1}{c_0+d_0}+\frac{3.a_1+a_0}{c_1+d_1}\\ +\frac{a_0+a_1}{c_0+d_1}+\frac{a_0+a_1}{c_1+d_0}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3.a_1{b}_1+a_0.b_0+a_1.b_0+a_0.b_1}{c_1+d_1}\\ -\frac{3.a_0{b}_0+a_1.b_1+a_1.b_0+a_0.b_1}{c_0+d_0}\\ +\frac{3.a_0{b}_0+a_1.b_1+a_1.b_0+a_0.b_1}{c_1+d_0}\\ -\frac{3.a_1{b}_1+a_0.b_0+a_1.b_0+a_0.b_1}{c_0+d_1}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3.a_1{b}_1+a_0.b_0+a_1.b_0+a_0.b_1}{c_1+d_1}\\ -\frac{3.a_0{b}_0+a_1.b_1+a_1.b_0+a_0.b_1}{c_0+d_0}\\ +\frac{3.a_0{b}_0+a_1.b_1+a_1.b_0+a_0.b_1}{c_0+d_1}\\ -\frac{3.a_1{b}_1+a_0.b_0+a_1.b_0+a_0.b_1}{c_1+d_0}\end{array}\right)$
$P=\frac{a \ast b}{c-d}$ ($\mathrm{c}-\mathrm{d}\ne 0)$	$\frac{\Delta a}{12}\left(\begin{array}{c}\frac{3.b_0+b_1}{c_0-d_0}+\frac{3.b_1+b_0}{c_1-d_1}\\ +\frac{b_0+b_1}{c_0-d_1}+\frac{b_0+b_1}{c_1-d_0}\end{array}\right)$	$\frac{\Delta b}{12}\left(\begin{array}{c}\frac{3.a_0+a_1}{c_0-d_0}+\frac{3.a_1+a_0}{c_1-d_1}\\ +\frac{a_0+a_1}{c_0-d_1}+\frac{a_0+a_1}{c_1-d_0}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3.a_1{b}_1+a_0.b_0+a_1.b_0+a_0.b_1}{c_1-d_1}\\ -\frac{3.a_0{b}_0+a_1.b_1+a_1.b_0+a_0.b_1}{c_0-d_0}\\ +\frac{3.a_0{b}_0+a_1.b_1+a_1.b_0+a_0.b_1}{c_1-d_0}\\ -\frac{3.a_1{b}_1+a_0.b_0+a_1.b_0+a_0.b_1}{c_0-d_1}\end{array}\right)$	$\frac{1}{12}\left(\begin{array}{c}\frac{3.a_1{b}_1+a_0.b_0+a_1.b_0+a_0.b_1}{c_1-d_1}\\ -\frac{3.a_0{b}_0+a_1.b_1+a_1.b_0+a_0.b_1}{c_0-d_0}\\ +\frac{3.a_0{b}_0+a_1.b_1+a_1.b_0+a_0.b_1}{c_0-d_1}\\ -\frac{3.a_1{b}_1+a_0.b_0+a_1.b_0+a_0.b_1}{c_1-d_0}\end{array}\right)$
$P=\frac{a\left(b+c\right)}{d}$ ($d\ne 0)$	$\frac{\Delta a}{6}\left[\begin{array}{c}\frac{2\left(b_0+c_0\right)+b_1+c_1}{d_0}\\ +\frac{2\left(b_1+c_1\right)+b_0+c_0}{d_1}\end{array}\right]$	$\frac{\Delta b}{6}\left[\begin{array}{c}\frac{2a_0+a_1}{d_0}\\ +\frac{2a_1+a_0}{d_1}\end{array}\right]$	$\frac{\Delta c}{6}\left[\begin{array}{c}\frac{2a_0+a_1}{d_0}\\ +\frac{2a_1+a_0}{d_1}\end{array}\right]$	$\frac{\left[\begin{array}{c}\left(2a_0+a_1\right)\left(b_0+c_0\right)\\ +\left(2a_1+a_0\right)\left(b_1+c_1\right)\end{array}\right]}{6}\left(\frac{1}{d_1}-\frac{1}{d_0}\right)$
$P=\frac{a\left(b-c\right)}{d}$ $\left(d\ne 0\right)$	$\frac{\Delta a}{6}\left[\begin{array}{c}\frac{2\left(b_0-c_0\right)+b_1-c_1}{d_0}\\ +\frac{2\left(b_1-c_1\right)+b_0-c_0}{d_1}\end{array}\right]$	$\frac{\Delta b}{6}\left[\begin{array}{c}\frac{2a_0+a_1}{d_0}\\ +\frac{2a_1+a_0}{d_1}\end{array}\right]$	$\frac{-\Delta c}{6}\left[\begin{array}{c}\frac{2a_0+a_1}{d_0}\\ +\frac{2a_1+a_0}{d_1}\end{array}\right]$	$\frac{\left[\begin{array}{c}\left(2a_0+a_1\right)\left(b_0-c_0\right)\\ +\left(2a_1+a_0\right)\left(b_1-c_1\right)\end{array}\right]}{6}\left(\frac{1}{d_1}-\frac{1}{d_0}\right)$
$P=\frac{a}{b\left(c+d\right)}$ ($b\ne 0;$ $c+d\ne 0$)	$\frac{\Delta a}{12}\left[\begin{array}{c}\frac{3}{b_0\left(c_0+d_0\right)}+\frac{3}{b_1\left(c_1+d_1\right)}\\ +\frac{1}{b_1\left(c_0+d_0\right)}+\frac{1}{b_1\left(c_1+d_0\right)}\\ +\frac{1}{b_1\left(c_0+d_1\right)}+\frac{1}{b_0\left(c_1+d_1\right)}\\ +\frac{1}{b_0\left(c_1+d_0\right)}+\frac{1}{b_0\left(c_0+d_1\right)}\end{array}\right]$	$\frac{1}{12}\left[\begin{array}{c}\frac{3a_1+a_0}{b_1\left(c_1+d_1\right)}-\frac{3a_1+a_0}{b_0\left(c_1+d_1\right)}\\ +\frac{3a_0+a_1}{b_1\left(c_0+d_0\right)}-\frac{3a_0+a_1}{b_0\left(c_0+d_0\right)}\\ +\frac{a_0+a_1}{b_1\left(c_1+d_0\right)}-\frac{a_0+a_1}{b_0\left(c_1+d_0\right)}\\ +\frac{a_0+a_1}{b_1\left(c_0+d_1\right)}-\frac{a_0+a_1}{b_0\left(c_0+d_1\right)}\end{array}\right]$	$\frac{1}{12}\left[\begin{array}{c}\frac{3a_1+a_0}{b_1\left(c_1+d_1\right)}-\frac{3a_1+a_0}{b_1\left(c_0+d_1\right)}\\ +\frac{3a_0+a_1}{b_0\left(c_1+d_0\right)}-\frac{3a_0+a_1}{b_0\left(c_0+d_0\right)}\\ +\frac{a_0+a_1}{b_1\left(c_1+d_0\right)}-\frac{a_0+a_1}{b_1\left(c_0+d_0\right)}\\ +\frac{a_0+a_1}{b_0\left(c_1+d_1\right)}-\frac{a_0+a_1}{b_0\left(c_0+d_1\right)}\end{array}\right]$	$\frac{1}{12}\left[\begin{array}{c}\frac{3a_1+a_0}{b_1\left(c_1+d_1\right)}-\frac{3a_1+a_0}{b_1\left(c_1+d_0\right)}\\ +\frac{3a_0+a_1}{b_0\left(c_0+d_1\right)}-\frac{3a_0+a_1}{b_0\left(c_0+d_0\right)}\\ +\frac{a_0+a_1}{b_1\left(c_0+d_1\right)}-\frac{a_0+a_1}{b_1\left(c_0+d_0\right)}\\ +\frac{a_0+a_1}{b_0\left(c_1+d_1\right)}-\frac{a_0+a_1}{b_0\left(c_1+d_0\right)}\end{array}\right]$
$P=\frac{a}{b\left(c-d\right)}$ ($b\ne 0;$ $c-d\ne 0$ )	$\frac{\Delta a}{12}\left[\begin{array}{c}\frac{3}{b_0\left(c_0-d_0\right)}+\frac{3}{b_1\left(c_1-d_1\right)}\\ +\frac{1}{b_1\left(c_0-d_0\right)}+\frac{1}{b_1\left(c_1-d_0\right)}\\ +\frac{1}{b_1\left(c_0-d_1\right)}+\frac{1}{b_0\left(c_1-d_1\right)}\\ +\frac{1}{b_0\left(c_1-d_0\right)}+\frac{1}{b_0\left(c_0-d_1\right)}\end{array}\right]$	$\frac{1}{12}\left[\begin{array}{c}\frac{3a_1+a_0}{b_1\left(c_1-d_1\right)}-\frac{3a_1+a_0}{b_0\left(c_1-d_1\right)}\\ +\frac{3a_0+a_1}{b_1\left(c_0-d_0\right)}-\frac{3a_0+a_1}{b_0\left(c_0-d_0\right)}\\ +\frac{a_0+a_1}{b_1\left(c_1-d_0\right)}-\frac{a_0+a_1}{b_0\left(c_1-d_0\right)}\\ +\frac{a_0+a_1}{b_1\left(c_0-d_1\right)}-\frac{a_0+a_1}{b_0\left(c_0-d_1\right)}\end{array}\right]$	$\frac{1}{12}\left[\begin{array}{c}\frac{3a_1+a_0}{b_1\left(c_1-d_1\right)}-\frac{3a_1+a_0}{b_1\left(c_0-d_1\right)}\\ +\frac{3a_0+a_1}{b_0\left(c_1-d_0\right)}-\frac{3a_0+a_1}{b_0\left(c_0-d_0\right)}\\ +\frac{a_0+a_1}{b_1\left(c_1-d_0\right)}-\frac{a_0+a_1}{b_1\left(c_0-d_0\right)}\\ +\frac{a_0+a_1}{b_0\left(c_1-d_1\right)}-\frac{a_0+a_1}{b_0\left(c_0-d_1\right)}\end{array}\right]$	$\frac{1}{12}\left[\begin{array}{c}\frac{3a_1+a_0}{b_1\left(c_1-d_1\right)}-\frac{3a_1+a_0}{b_1\left(c_1-d_0\right)}\\ +\frac{3a_0+a_1}{b_0\left(c_0-d_1\right)}-\frac{3a_0+a_1}{b_0\left(c_0-d_0\right)}\\ +\frac{a_0+a_1}{b_1\left(c_0-d_1\right)}-\frac{a_0+a_1}{b_1\left(c_0-d_0\right)}\\ +\frac{a_0+a_1}{b_0\left(c_1-d_1\right)}-\frac{a_0+a_1}{b_0\left(c_1-d_0\right)}\end{array}\right]$

present the derived mathematical expressions and constraints for determining the influences of the involved factors after the averaged chain substitution method containing up to four factor variables, respectively, for the: Multiplicative and Multiple Factor Models; Additive or Different-Multiple Factor Models; Multiplicative-Multiple Factor Models; Additive or Different-Multiple Factor Models; Additive or Different-Multiplicative-Multiple Factor Models. There are restrictions only with the multiple and mixed models containing an element of the multiple model. The constraints are presented in Column One below the mathematical entry of the factor model.

When determining the mathematical expressions for quantifying the individual factor influences in the three- and four-multiple factor models, the factor model should be presented in a simplified form of a multiplicative-multiple form, as shown in the three final rows of the first column in Table 3. Otherwise, the direct application of the averaged chain substitution method results in deriving incorrect mathematical expressions for determining the individual factor influences.

The approbation of the methodology of the averaged chain substitution method with assigning quantitative values of the basic (planned) and actual (current) values of the factor variables has been carried out in the MS Excel Version 2406 environment. A multitude of combinations of the values of the factor variables were used in the testing for approval in order to confirm the accuracy of the results obtained from the derived mathematical expressions for quantifying the individual factor influences of the factor models presented in Table 3, Table 4, Table 5, Table 6 and Table 7.

According to Mitev [21] (p. 97): “For those factor models, for which the scientific literature offers analytical expressions developed for the purposes of determining the individual factor influences after the integral method, namely: $P=a \ast b$, $P=a \ast b \ast c$ and $P=a \ast b \ast c \ast d$ quantitative results have been obtained that are identical to those after the averaged chain substitution method. Only in the case of the additive-multiple factor models of the type $P=\frac{a}{b}$, $P=\frac{a}{b+c}$ and $P=\frac{a}{b+c+d}$, deviations in the quantitative values of the individual factor influences after the two methods have been obtained”. The obtained relative deviations for the quantitative values of the individual factor influences after the two methods vary from 3 to 250%. The largest relative deviation (error) occurs with a two-factor multiple model, where the deviation of factor b, which is in the denominator, reaches and even exceeds 250% when its absolute deviation ($\Delta b$) increases significantly and, accordingly, $\left|\frac{b_1}{b_0}\right|$ also rises.

The results of the approbation of the averaged chain substitution method confirm the universality of the method regarding the type of factor model and the accuracy which characterizes it when directly quantifying the individual factor influences from the change of the factor variables on the change of the performance indicator in the factor models presented.

As can be seen from Table 3, Table 4, Table 5, Table 6 and Table 7, for factor models with more than two factor variables, the derived mathematical expressions for determining the individual factor effects after the averaged chain substitution method are significantly more complicated, especially when the factor models are mixed, containing a multiple element with more factor variables in the denominator. Therefore, we can claim that with the increase in the number of factor variables, the mathematical expressions for determining individual factor influences also become complicated. This drawback can easily be overcome by using pre-developed templates in spreadsheets or in the MS Excel environment.

5. Derivation of Individual Factor Influences for Five-Factor Models

If we look at the derived mathematical expressions for individual factor influences in additive or difference-multiplicative factor models using the averaged chain substitution method (Table 4), using the analogy method we can easily derive similar formulas for five or more factor additive or difference-multiplicative factor models. Mathematical expressions for determining individual factor influences in five-factor additive or difference-multiplicative factor models are presented in

Table 8. Mathematical expressions for determining the influence of individual factors in five-factor additive or difference-multiplicative factor models.

Factor Models, (Limitation)	$\mathbf{Factor} \mathbf{Influence} \mathit{a},$ $\Delta {\mathit{P}}_{\left(\mathit{a}\right)}$	$\mathbf{Factor} \mathbf{Influence} \mathit{b},$ $\Delta {\mathit{P}}_{\left(\mathit{b}\right)}$	$\mathbf{Factor} \mathbf{Influence} \mathit{c},$ $\Delta {\mathit{P}}_{\left(\mathit{c}\right)}$	$\mathbf{Factor} \mathbf{Influence} \mathit{d},$ $\Delta {\mathit{P}}_{\left(\mathit{d}\right)}$	$\mathbf{Factor} \mathbf{Influence} \mathit{e},$ $\Delta {\mathit{P}}_{\left(\mathit{e}\right)}$
Additive or difference-multiplicative factor models
$P=a\left(b+c+d+e\right)$	$\frac{\Delta a}{2}\left(\begin{array}{c}{b}_0+b_1+c_0+c_1\\ +d_0+d_1+e_0+e_1\end{array}\right)$	$\frac{\Delta b}{2}\left(a_0+a_1\right)$	$\frac{\Delta c}{2}\left(a_0+a_1\right)$	$\frac{\Delta d}{2}\left(a_0+a_1\right)$	$\frac{\Delta e}{2}\left(a_0+a_1\right)$
$P=a\left(b-c-d-e\right)$	$\frac{\Delta a}{2}\left(\begin{array}{c}{b}_0+b_1-c_0-c_1\\ -d_0-d_1-e_0-e_1\end{array}\right)$	$\frac{\Delta b}{2}\left(a_0+a_1\right)$	$\frac{-\Delta c}{2}\left(a_0+a_1\right)$	$\frac{-\Delta d}{2}\left(a_0+a_1\right)$	$\frac{-\Delta e}{2}\left(a_0+a_1\right)$
$P=a\left(b+c-d-e\right)$	$\frac{\Delta a}{2}\left(\begin{array}{c}{b}_0+b_1+c_0+c_1\\ -d_0-d_1-e_0-e_1\end{array}\right)$	$\frac{\Delta b}{2}\left(a_0+a_1\right)$	$\frac{\Delta c}{2}\left(a_0+a_1\right)$	$\frac{-\Delta d}{2}\left(a_0+a_1\right)$	$\frac{-\Delta e}{2}\left(a_0+a_1\right)$
$P=a\left(b+c+d-e\right)$	$\frac{\Delta a}{2}\left(\begin{array}{c}{b}_0+b_1+c_0+c_1\\ +d_0+d_1-e_0-e_1\end{array}\right)$	$\frac{\Delta b}{2}\left(a_0+a_1\right)$	$\frac{\Delta c}{2}\left(a_0+a_1\right)$	$\frac{\Delta d}{2}\left(a_0+a_1\right)$	$\frac{-\Delta e}{2}\left(a_0+a_1\right)$
$P=a\left(b-c+d-e\right)$	$\frac{\Delta a}{2}\left(\begin{array}{c}{b}_0+b_1-c_0-c_1\\ +d_0+d_1-e_0-e_1\end{array}\right)$	$\frac{\Delta b}{2}\left(a_0+a_1\right)$	$\frac{-\Delta c}{2}\left(a_0+a_1\right)$	$\frac{\Delta d}{2}\left(a_0+a_1\right)$	$\frac{-\Delta e}{2}\left(a_0+a_1\right)$
$P=a\left(b-c+d+e\right)$	$\frac{\Delta a}{2}\left(\begin{array}{c}{b}_0+b_1-c_0-c_1\\ +d_0+d_1+e_0+e_1\end{array}\right)$	$\frac{\Delta b}{2}\left(a_0+a_1\right)$	$\frac{-\Delta c}{2}\left(a_0+a_1\right)$	$\frac{\Delta d}{2}\left(a_0+a_1\right)$	$\frac{\Delta e}{2}\left(a_0+a_1\right)$
$P=a\left(b-c-d+e\right)$	$\frac{\Delta a}{2}\left(\begin{array}{c}{b}_0+b_1-c_0-c_1\\ -d_1+e_0+e_1\end{array}\right)$	$\frac{\Delta b}{2}\left(a_0+a_1\right)$	$\frac{\Delta c}{2}\left(a_0+a_1\right)$	$\frac{-\Delta d}{2}\left(a_0+a_1\right)$	$\frac{\Delta e}{2}\left(a_0+a_1\right)$

Source: author’s development.

.

In a similar way, based on mathematical expressions for individual factor influences for two-, three- and four-factor additive or difference-multiple models with an additive or difference part in the numerator of the factor model, presented in Table 6, using the method of analogy we can derive factor influences for five or more factor models of the same type. Mathematical expressions for determining individual factor influences in five-factor additive or difference-multiple models with an additive or difference part in the numerator of the factor model are presented in

Table 9. Mathematical expressions for determining the influences of individual factors in five-factor additive or difference-multiple factor models.

Factor Models, (Limitation)	$\mathbf{Factor} \mathbf{Influence} \mathit{a},$ $\Delta {\mathit{P}}_{\left(\mathit{a}\right)}$	$\mathbf{Factor} \mathbf{Influence} \mathit{b},$ $\Delta {\mathit{P}}_{\left(\mathit{b}\right)}$	$\mathbf{Factor} \mathbf{Influence} \mathit{c},$ $\Delta {\mathit{P}}_{\left(\mathit{c}\right)}$	$\mathbf{Factor} \mathbf{Influence} \mathit{d},$ $\Delta {\mathit{P}}_{\left(\mathit{d}\right)}$	$\mathbf{Factor} \mathbf{Influence} \mathit{e},$ $\Delta {\mathit{P}}_{\left(\mathit{e}\right)}$
Additive or difference-multiple factor models
$\begin{array}{c}P=\frac{a+b+c+d}{e}\\ \left(e\ne 0\right)\end{array}$	$\frac{\Delta a}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{\Delta b}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{\Delta c}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{\Delta d}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{\begin{array}{c}{a}_0+a_1+b_0+b_1\\ +c_0+c_1+d_0+d_1\end{array}}{2}\left(\frac{1}{e_1}-\frac{1}{e_0}\right)$
$\begin{array}{c}P=\frac{a-b-c-d}{e}\\ \left(e\ne 0\right)\end{array}$	$\frac{\Delta a}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{-\Delta b}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{-\Delta c}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{-\Delta d}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{\begin{array}{c}{a}_0+a_1-b_0-b_1\\ -c_0-c_1-d_0-d_1\end{array}}{2}\left(\frac{1}{e_1}-\frac{1}{e_0}\right)$
$\begin{array}{c}P=\frac{a+b-c-d}{e}\\ \left(e\ne 0\right)\end{array}$	$\frac{\Delta a}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{\Delta b}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{-\Delta c}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{-\Delta d}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{\begin{array}{c}{a}_0+a_1+b_0+b_1\\ -c_0-c_1-d_0-d_1\end{array}}{2}\left(\frac{1}{e_1}-\frac{1}{e_0}\right)$
$\begin{array}{c}P=\frac{a+b+c-d}{e}\\ \left(e\ne 0\right)\end{array}$	$\frac{\Delta a}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{\Delta b}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{\Delta c}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{-\Delta d}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{\begin{array}{c}{a}_0+a_1+b_0+b_1\\ +c_0+c_1-d_0-d_1\end{array}}{2}\left(\frac{1}{e_1}-\frac{1}{e_0}\right)$
$\begin{array}{c}P=\frac{a-b+c-d}{e}\\ \left(e\ne 0\right)\end{array}$	$\frac{\Delta a}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{-\Delta b}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{\Delta c}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{-\Delta d}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{\begin{array}{c}{a}_0+a_1-b_0-b_1\\ +c_0+c_1-d_0-d_1\end{array}}{2}\left(\frac{1}{e_1}-\frac{1}{e_0}\right)$
$\begin{array}{c}P=\frac{a-b+c+d}{e}\\ \left(e\ne 0\right)\end{array}$	$\frac{\Delta a}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{-\Delta b}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{\Delta c}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{\Delta d}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{\begin{array}{c}{a}_0+a_1-b_0-b_1\\ +c_0+c_1+d_0+d_1\end{array}}{2}\left(\frac{1}{e_1}-\frac{1}{e_0}\right)$
$\begin{array}{c}P=\frac{a-b-c+d}{e}\\ \left(e\ne 0\right)\end{array}$	$\frac{\Delta a}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{-\Delta b}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{-\Delta c}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{\Delta d}{2}\left(\frac{1}{e_0}+\frac{1}{e_1}\right)$	$\frac{\begin{array}{c}{a}_0+a_1-b_0-b_1\\ -c_0-c_1+d_0+d_1\end{array}}{2}\left(\frac{1}{e_1}-\frac{1}{e_0}\right)$

Source: author’s development.

.

Derived mathematical expressions for determining individual factor influences in five-factor models presented in Table 8 and Table 9, certified and confirmed in MS Excel. The test on which the certification of the derived formulas is based is that the sum of the individual factor influences, namely the complex factor influence, must be equal to the absolute change in the resultative indicator. During certification, many combinations of input values of factor variables were used to confirm the accuracy of the derived mathematical expressions for determining individual factor influences presented in Table 8 and Table 9 factor models.

6. Dynamic DFA by the Averaged Chain Substitution Method—Examples

The destabilized form of the total liquidity ratio ($TLR$) has the following deterministic factor model:

$$TL{R}_t=\frac{M{I}_t+C{R}_t+CF{A}_t+CC{E}_t}{C{L}_t}$$

(29)

where: $M{I}_t$ is the value of material inventories for the period $t$, in BGN thousands;

• $C{R}_t$—the value of current receivables for the period $t$, in BGN thousands;
• $CF{A}_t$—the value of current financial assets for the period $t$, in BGN thousands;
• $CC{E}_t$—the value of cash and cash equivalents for the period $t$, in BGN thousands;
• $C{L}_t$—the value of current liabilities for the period $t$, in BGN thousands;
• $t$—the index of the $t$th value of the performance indicator and of the participating factor variables over time, $t=0,1,2,\dots ,T$.

Mathematical expressions for determining the individual factor influences of a five-factor additive-multiple model of the total liquidity ratio, presented in Table 7, are transformed into the following form:

• for a factor value of inventories $\left(In\right)$:

  $$\Delta TLR{\left(MI\right)}_t=\frac{\Delta MI}{2}\left(\frac{1}{C{L}_{t-1}}+\frac{1}{C{L}_t}\right)$$

  (30)
• for a factor value of short-term receivables $\left(CR\right)$:

  $$\Delta TLR{\left(CR\right)}_t=\frac{\Delta CR}{2}\left(\frac{1}{C{L}_{t-1}}+\frac{1}{C{L}_t}\right)$$

  (31)
• for a factor value of current financial assets $\left(CFA\right)$:

  $$\Delta TLR{\left(CFA\right)}_t=\frac{\Delta CFA}{2}\left(\frac{1}{C{L}_{t-1}}+\frac{1}{C{L}_t}\right)$$

  (32)
• for a factor value of cash and cash equivalents $\left(CCE\right)$:

  $$\Delta TLR{\left(CCE\right)}_t=\frac{\Delta CCE}{2}\left(\frac{1}{C{L}_{t-1}}+\frac{1}{C{L}_t}\right)$$

  (33)
• for a factor value of current liabilities $\left(CL\right)$:

  $$\Delta TLR{\left(CL\right)}_t=\frac{M{I}_{t-1}+M{I}_t+C{R}_{t-1}+C{R}_t+CF{A}_{t-1}+CF{A}_t+CC{E}_{t-1}+CC{E}_t}{2}\left(\frac{1}{C{L}_t}-\frac{1}{C{L}_{t-1}}\right)$$

  (34)

The necessary data for performing a dynamic deterministic factor analysis of the total liquidity of Monbat JSC, Bulgaria and ELHIM-ISKRA JSC, Bulgaria for the period 2017–2021 are freely accessible and are taken from the consolidated annual financial statements of the enterprises, available on the website of the Bulgarian Stock Exchange (https://www.bse-sofia.bg/bg/disclosure-by-issuers) accessed on 12 April 2024.

The input data and the results obtained from the deterministic factor analysis of the total liquidity ratio of Monbat JSC for the period 2017–2021 are presented in

Table 10. Input data and results obtained from the deterministic factor analysis of the total liquidity ratio of Monbat JSC for the period 2017–2022.

Input Data
Indicator	Period
	2017	2018	2019	2020	2021	2022
Value of Material Inventories, (MI), BGN’000	80,633	85,104	97,926	99,269	104,761	105,398
Value of Current Receivables, (CR), BGN’000	120,193	130,990	122,667	113,002	132,411	137,451
Value of Current Financial Assets, (CFA), BGN’000	23,581	25,499	20	157	132	33,816
Value of Cash and Cash Equivalents, (CCE), BGN’000	7472	37,308	23,913	24,008	9025	8137
Value of current liabilities, (CL), BGN’000	32,184	36,575	43,022	35,694	45,085	44,822
Total Liquidity Ratio, (TLR)	7.2048	7.6255	5.6837	6.6240	5.4637	6.3541
Results obtained
Indicator	Analyzed period
	2017–2018	2018–2019	2019–2020	2020–2021	2021–2022	2017–2022
	1	2	3	4	5	0–5
Absolute change in material inventories, (ΔMI = MIt − MIt−1), BGN’000	4471	12,822	1343	5492	637	24,765
Relative change in material inventories, (%MI = ΔMI * 100/MIt−1), %	5.54%	15.07%	1.37%	5.53%	0.61%	30.71%
Absolute change in current receivables, (ΔCR = CRt − CRt−1), BGN’000	10,797	−8323	−9665	19,409	5040	17,258
Relative change in current receivables, (%CR = ΔCR * 100/CRt−1), %	8.98%	−6.35%	−7.88%	17.18%	3.81%	14.36%
Absolute change in current financial assets, (ΔCF = CFt − CFt−1), BGN’000	1918	−25,479	137	−25	33,684	10,235
Relative change in current financial assets (%CF = ΔCF * 100/CFt−1), %	8.13%	−99.92%	685.00%	−15.92%	25518.18%	43.40%
Absolute change in cash and cash equivalents (ΔCCE = CCEt − CCEt−1), BGN’000	29,836	−13,395	95	−14,983	−888	665
Relative change in cash and cash equivalents (%CF = ΔCF * 100/CFt − 1), %	399.30%	−35.90%	0.40%	−62.41%	−9.84%	8.90%
Absolute change in current liabilities (ΔCL = CLt − CLt−1), BGN’000	4391	6447	−7328	9391	−263	12,638
Relative change in current liabilities (%CL = ΔCL * 100/CLt−1), %	13.64%	17.63%	−17.03%	26.31%	−0.58%	39.27%
Absolute change in total liquidity ratio (ΔTL = TLt − TLt−1)	0.4207	−1.9417	0.9402	−1.1603	0.8904	−0.8507
Relative change in total liquidity ratio (%TL = ΔTL * 100/TLt−1), %	5.84%	−25.46%	16.54%	−17.52%	16.30%	−11.81%
Quantitative impact of material inventories (ΔTLR(In))	0.1306	0.3243	0.0344	0.1378	0.0142	0.6610
Relative influence of material inventories (%TLR(MI) = ΔTLR(MI) * 100/TLRt−1), %	1.81%	4.25%	0.61%	2.08%	0.26%	9.17%
Quantitative impact of current receivables (ΔTLR(CR))	0.3153	−0.2105	−0.2477	0.4871	0.1121	0.4606
Relative impact of current receivables (%TLR(CR) = ΔTLR(CR) * 100/TLRt−1), %	4.38%	−2.76%	−4.36%	7.35%	2.05%	6.39%
Quantitative impact of current financial assets (ΔTLR(CFA))	0.0560	−0.6444	0.0035	−0.0006	0.7493	0.2732
Relative influence of current financial assets (%TLR(CFA) = ΔTLR(CFA) * 100/TLRt−1), %	0.78%	−8.45%	0.06%	−0.01%	13.71%	3.79%
Quantitative impact of cash and cash equivalents (ΔTLR(CCE))	0.8714	−0.3388	0.0024	−0.3760	−0.0198	0.0177
Relative influence of cash and cash equivalents (%TLR(CCE) = ΔTLR(CCE) * 100/TRLt−1), %	12.09%	−4.44%	0.04%	−5.68%	−0.36%	0.25%
Quantitative impact of current liabilities (ΔTLR(CL))	−0.9527	−1.0723	1.1476	−1.4086	0.0346	−2.2633
Relative impact of current liabilities (%TLR(CL) = ΔTLR(CL) * 100/TLRt−1), %	13.22%	14.06%	−20.19%	21.27%	−0.63%	31.41%
Complex influence ΔTLR = ΔTLR(MI) + ΔTLR(CR) +ΔTLR(CFA) + ΔTLR(CCE) + ΔTLR(CL)	0.4207	−1.9417	0.9402	−1.1603	0.8904	−0.8507
Cumulative relative influence %TLR = %TLR(MI) + %TLR(CR) + %TLR(CFA) + %TLR(CCE) + %TLR(CL), %	5.84%	−25.46%	16.54%	−17.52%	16.30%	−11.81%
Verification: ΔTLR = ΔTLR(MI) + ΔTLR(CR) + ΔTLR(CFA) + ΔTLR(CCE) + ΔTLR(CL)	False	True	True	True	False	True
Value of the absolute error, STLR, BGN/BGN	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
Verification: %TLR = %TLR(MI) + %TLR(CR) + %TLR(CFA) + %TLR(CCE) + %TLR(CL)	False	False	False	False	False	False
Value of the relative error, δTLR, %	0.00000%	0.00000%	0.00000%	0.00000%	0.00000%	0.00000%

and

Table 11. Input data and results obtained from the deterministic factor analysis of the total liquidity ratio of ELHIM-ISKRA JSC for the period 2017–2022.

Input Data
Indicator	Period
	2017	2018	2019	2020	2021	2022
Value of Material Inventories, (MI), BGN’000	10,392	10,728	12,058	10,251	10,361	11,565
Value of Current Receivables, (CR), BGN’000	6926	5264	3195	4009	4578	4989
Value of Current Financial Assets, (CFA), BGN’000	0	0	0	0	0	0
Value of Cash and Cash Equivalents, (CCE), BGN’000	783	1080	1841	3497	3330	2517
Value of current liabilities, (CL), BGN’000	1531	1780	1481	1345	1920	2239
Total Liquidity Ratio, (TLR)	11.8230	9.5910	11.5422	13.2022	9.5151	8.5176
Results obtained
Indicator	Analyzed period
	2017–2018	2018–2019	2019–2020	2020–2021	2021–2022	2017–2022
	1	2	3	4	5	0–5
Absolute change in material inventories, (ΔMI = MIt − MIt−1), BGN’000	336	1330	−1807	110	1204	1173
Relative change in material inventories, (%MI = ΔMI * 100/MIt−1), %	3.23%	12.40%	−14.99%	1.07%	11.62%	11.29%
Absolute change in current receivables, (ΔCR = CRt − CRt−1), BGN’000	−1662	−2069	814	569	411	−1937
Relative change in current receivables, (%CR = ΔCR * 100/CRt−1), %	−24.00%	−39.30%	25.48%	14.19%	8.98%	−27.97%
Absolute change in current financial assets, (ΔCF = CFt − CFt−1), BGN’000	0	0	0	0	0	0
Relative change in current financial assets, (%CF = ΔCF * 100/CFt−1), %	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%
Absolute change in cash and cash equivalents, (ΔCCE = CCEt − CCEt−1), BGN’000	297	761	1656	−167	−813	1734
Relative change in cash and cash equivalents, (%CF = ΔCF * 100/CFt − 1), %	37.93%	70.46%	89.95%	−4.78%	−24.41%	221.46%
Absolute change in current liabilities, (ΔCL = CLt − CLt−1), BGN’000	249	−299	−136	575	319	708
Relative change in current liabilities, (%CL = ΔCL * 100/CLt−1), %	16.26%	−16.80%	−9.18%	42.75%	16.61%	46.24%
Absolute change in total liquidity ratio, (ΔTL = TLt − TLt−1)	−2.2320	1.9512	1.6600	−3.6871	−0.9975	−3.3053
Relative change in total liquidity ratio, (%TL = ΔTL * 100/TLt−1), %	−18.88%	20.34%	14.38%	−27.93%	−10.48%	−27.96%
Quantitative impact of material inventories, (ΔTLR(In))	0.2041	0.8226	−1.2818	0.0695	0.5824	0.6450
Relative influence of material inventories, (%TLR(MI) = ΔTLR(MI) * 100/TLRt−1), %	1.73%	8.58%	−11.11%	0.53%	6.12%	5.46%
Quantitative impact of current receivables, (ΔTLR(CR))	−1.0096	−1.2797	0.5774	0.3597	0.1988	−1.0652
Relative impact of current receivables, (%TLR(CR) = ΔTLR(CR) * 100/TLRt−1), %	−8.54%	−13.34%	5.00%	2.72%	2.09%	−9.01%
Quantitative impact of current financial assets (ΔTLR(CFA))	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
Relative influence of current financial assets, (%TLR(CFA) = ΔTLR(CFA) * 100/TLRt−1), %	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%
Quantitative impact of cash and cash equivalents, (ΔTLR(CCE))	0.1804	0.4707	1.1747	−0.1056	−0.3933	0.9535
Relative influence of cash and cash equivalents, (%TLR(CCE) = ΔTLR(CCE) * 100/TRLt−1), %	1.53%	4.91%	10.18%	−0.80%	−4.13%	8.06%
Quantitative impact of current liabilities, (ΔTLR(CL))	−1.6069	1.9376	1.1897	−4.0108	−1.3854	−3.8388
Relative impact of current liabilities, (%TLR(CL) = ΔTLR(CL) * 100/TLRt−1), %	13.59%	−20.20%	−10.31%	30.38%	14.56%	32.47%
Complex influence, ΔTLR = ΔTLR(MI) + ΔTLR(CR) + ΔTLR(CFA) + ΔTLR(CCE) + ΔTLR(CL)	−2.2320	1.9512	1.6600	−3.6871	−0.9975	−3.3053
Cumulative relative influence, %TLR = %TLR(MI) + %TLR(CR) + %TLR(CFA) + %TLR(CCE) + %TLR(CL), %	−18.88%	20.34%	14.38%	−27.93%	−10.48%	−27.96%
Verification: ΔTLR = = ΔTLR(MI) + ΔTLR(CR) + ΔTLR(CFA) + ΔTLR(CCE) + ΔTLR(CL)	True	True	True	True	True	True
Value of the absolute error, STLR, BGN/BGN	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
Verification: %TLR = = %TLR(MI) + %TLR(CR) + %TLR(CFA) + %TLR(CCE) + %TLR(CL)	False	False	False	False	False	False
Value of the relative error, δTLR, %	0.00000%	0.00000%	0.00000%	0.00000%	0.00000%	0.00000%

.

Verification of the obtained results is mandatory and includes comparing the absolute change of the resultative indicator to the sum of the five factor influences. Verification of the accuracy of the obtained results shows insignificant absolute and relative errors. The maximum absolute error does not exceed 0.0000001 and the maximum relative error does not exceed 0.000001%. This confirms the extremely high accuracy with which the averaged chain substitution method, implemented in MS Excel, is characterized.

Figure 2 presents the quantitative impact of absolute changes of the variable factors of value of material inventories, value of current receivables, value of financial assets, value of cash and cash equivalents, and value of current liabilities on the absolute change of the resultative indicator—total liquidity ratio by sub-period.

Figure 3 presents the quantitative impact of absolute changes of the variable factors of value of material inventories, value of current receivables, value of financial assets, value of cash and cash equivalents, and value of current liabilities on the absolute change of the resultative indicator—total liquidity ratio by sub-period of ELHIM-ISKRA JSC.

The results of the presented dynamic DFA of the detailed form of the indicators of the total liquidity of Monbat JSC and ELHIM-ISKRA JSC for the period 2017–2022 unconditionally emphasize the easy applicability and universality of the averaged method of chain substitutions for the purposes of the dynamic DFA of economic, financial, or other mathematically determined indicators. The averaged chain substitution method is the only accurate method of deterministic factor analysis that is applicable to deterministic factor models containing a multiple element in them.

To avoid calculation errors, it is mandatory to check the accuracy of the obtained absolute and relative factor influences of the participating factor variables, respectively, on the absolute and relative change of the resultative indicator.

In the dynamic DFA by the averaged chain substitution method, the absolute and relative errors tend to zero. This confirms the accuracy of the averaged chain substitution method.

7. Discussion

The existing DFA methods are characterized by a number of advantages, but also by many disadvantages, and also by a different degree of universality (applicability), i.e., most of them have been derived only for a limited number and types of factorial models.

To date, there is no other unified DFA method that is universal, i.e., applicable to all types of factorial models. An exception is the method of chain substitutions. However, accurate and unambiguous results regarding the quantification of factor influences cannot be obtained through it.

The presented new method of DFA, namely the averaged chain substitution method, solves a long-standing scientific and applied problem, namely accuracy and unambiguity regarding the quantification of individual factor influences on the change of the resultative indicator in deterministic factor models.

The averaged chain substitution method is characterized by full universality of application regarding all types of deterministic factor models, and through it, accurate and unambiguous mathematical expressions and results are obtained for the accurate and unambiguous quantification of the individual factor influences of the participating factor variables for all types of factor models.

In deterministic factor models containing a multiple element, the averaged chain substitution method is the only method that is characterized by absolute accuracy of the obtained results.

Dynamic deterministic factor analysis using the averaged chain substitution method enables fast, easy and accurate quantification of individual factor influences in a deterministic factor model. Through it, the trends in the development of resultative indicators and the quantitative influences of the participating factors on the quantitative change of the resultative indicator can be revealed.

The averaged chain substitution method is applicable to perform dynamic deterministic factor analyses of deterministic factor models of resultative indicators from all scientific fields where there are mathematically determined factorial dependencies between the resultative indicator and the participating factor variables.

The results of the dynamic deterministic factor analysis serve to quantify the cause-and-effect relationships, formulating reasonable conclusions and guidelines for improving the future economic activity of the enterprise by financial analysts.

The averaging method of chain substitutions is a method used in economic and statistical analyzes to measure the impact of various factors on the results of a given process or phenomenon. The method is based on successive substitution of the factors with their mean values in order to estimate the influence of each variable separately. The main applications of the method are related to: productivity analysis, as it is used to break down total productivity into various components, such as labor productivity, capital productivity, etc.; economic research, to obtain an assessment of the impact of various macroeconomic factors on GDP, inflation, unemployment, etc.; financial analyses, to support and analyze the financial results of enterprises, considering the impact of various financial indicators; marketing research, for the purpose of breaking down sales and evaluating the impact of various marketing strategies.

The main advantages of the considered method are as follows: accuracy of the analysis, as the method allows a detailed examination of the influence of individual factors; flexibility, because it can be applied in different areas and for different types of analyses; ease of implementation and application, as the procedure is relatively easy to understand and apply; uncovering hidden dependencies, because it helps to discover interrelationships between different factors that may go unnoticed by other methods of analysis.

On the other hand, this approach is also characterized by disadvantages, such as: linearity of the analysis, as it implies linear dependencies between the factors, which is not always realistic and corresponds to the actual situation; dependence on the quality of the data, because the results obtained are highly dependent on the accuracy and completeness of the data used; complexity when working with a large number of factors, because when the number of factors is large, the method can become laborious and complex to implement and interpret; limited application in non-linear models, as by its nature the method is not suitable for analyzes where interactions between factors are highly non-linear.

The averaging method of chain substitutions is a powerful tool for economic and statistical analysis that allows detailed examination of the influence of various factors. However, the method has its limitations related to linearity assumptions and dependence on data quality. Its use must be carefully considered and combined with other methods to obtain reliable results.

8. Conclusions

The derived mathematical expressions for a direct, unambiguous, and precise quantification of individual factor influences eases the application of the method by all economic and financial analysts for the purposes of DFA.

The averaged chain substitution method is characterized by the accuracy and unambiguity of the results obtained, as well as the complete universality of the type of factor models. To date, the method has been developed for 60 types of factor models, and the derived mathematical expressions for the direct, unambiguous, and accurate quantitative determination of individual factor influences are 239. All of them are presented in Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9. This makes it easy to apply the method to all economic and financial analyses for DFA purposes.

The increase in the number of factor variables in the factor model brings about a rise in the number of combinations of the order of substitution of the basic (planned) and actual (current) values of the factor variables when constructing factor chains after the chain substitution method. This significantly complicates, yet does not render practically impossible, the derivation of the mathematical expressions for quantifying the individual factor influences on the change of the performance indicator in five- and more-factor models. However, this is a very laborious process, and the resulting mathematical expressions are significantly complicated, especially when there are multiple elements in the factor model. This is the only, yet surmountable, shortcoming of the averaged chain substitution method.

The averaged chain substitution method is not a modification of the chain substitution method but employs it in the construction of the factor chains after the chain substitution method with the various combinations of the order of substitution of the factor variables and the subsequent derivation of the individual factor influences. The averaged chain substitution method is a quantitative method and can find an even wider application, namely in deterministic factor analysis of economic and non-economic indicators alike.

The trends for the future development of the averaged chain substitution method are as follows: approbation of the method for five- and higher-factor models; approbation of the method for deterministic factor models containing mathematical, trigonometric, or other functions.